Photonic crystal light trapping: Beyond 30% conversion efficiency for silicon photovoltaics

The power conversion efficiency of single-junction silicon solar cells has increased only by 1.5% despite extensive efforts over the past two decades. The current world-record efficiencies of silicon solar cells, within the 25%–26.7% range, fall well below the thermodynamic limit of 32.3%. We review the recent progress in photonic crystal light-trapping architectures poised to achieve 28%–31% conversion efficiency in flexible 3–20 μm-thick, single-junction crystalline-silicon solar cells. These photonic crystals utilize wave-interference based light-trapping, enabling solar absorption well beyond the Lambertian limit in the 300–1200 nm wavelength range. Using experimentally feasible doping profiles, carrier lifetimes, surface recombination velocities, and established Auger recombination losses, we review considerations leading to the prediction of 31% efficiency in a 15 μm-thick silicon photonic crystal cell with interdigitated back-contacts. This is beyond the conversion efficiency of any single-material photovoltaic device of any thickness.The power conversion efficiency of single-junction silicon solar cells has increased only by 1.5% despite extensive efforts over the past two decades. The current world-record efficiencies of silicon solar cells, within the 25%–26.7% range, fall well below the thermodynamic limit of 32.3%. We review the recent progress in photonic crystal light-trapping architectures poised to achieve 28%–31% conversion efficiency in flexible 3–20 μm-thick, single-junction crystalline-silicon solar cells. These photonic crystals utilize wave-interference based light-trapping, enabling solar absorption well beyond the Lambertian limit in the 300–1200 nm wavelength range. Using experimentally feasible doping profiles, carrier lifetimes, surface recombination velocities, and established Auger recombination losses, we review considerations leading to the prediction of 31% efficiency in a 15 μm-thick silicon photonic crystal cell with interdigitated back-contacts. This is beyond the conversion efficiency of any single-material...


I. INTRODUCTION
Sunlight striking the earth provides approximately 173 000 TW of continuous power, roughly 10 000 times more than all worldwide power consumption. The realization of high-efficiency, low-cost, light-weight, flexible, solar energy conversion materials is central to exploiting this limitless but largely untapped resource. Silicon is valuable due to its non-toxicity, abundance, and mature fabrication technology. However, a thin silicon slab is a relatively weak absorber of sunlight due to its indirect electronic bandgap. In order to improve light-capture, a silicon solar cell typically utilizes a textured top surface. Most of the conventional cell designs, with efficiencies higher than 24%, [1][2][3]6 employ textures that consider light as rays rather than waves. The current world-record, single-junction silicon solar cell with 165 μm thickness has a power conversion efficiency of 26.7%. 6,7 However, this falls well below the thermodynamic efficiency limit of 32.33% for a single-junction crystalline silicon (c-Si) cell at room temperature, under 1-sun illumination. 8 Practical considerations such as Auger recombination and defect-mediated charge carrier recombination in the bulk and surface of the cell, imperfect trapping of sunlight in the cell, and bandgap narrowing (BGN) of silicon invariably reduce the maximum attainable efficiency to below the thermodynamic limit. Based on the assumption that light is a ray, devoid of all wave-interference effects, a limiting efficiency based solely on Auger charge carrier recombination was proposed. 9 In this oversimplified ray-optics picture, assuming a Lambertian light-trapping limit, a maximum power conversion efficiency of 29.43% was suggested. As a solar cell is made thicker, it absorbs more light but at the same time suffers from increased bulk-recombination. These two competing effects result in an optimum thickness of the cell. Lambertian ray-optics light-trapping 10 and an improved Auger recombination model 11 suggest an optimum thickness of 110 μm for an undoped, hypothetical cell with no other forms of non-radiative charge carrier recombination. 9 For cells that absorb sunlight below the Lambertian limit, the optimum thickness may be considerably greater and the maximum theoretical efficiency considerably less. Following this model, conventional solar cells are typically made hundreds of micrometer thick.
For example, the 25% efficient passivated emitter rear locally diffused (PERL) cell, the polycrystalline silicon on oxide (POLO) cell with 26.1% efficiency, and the current world-record holding Kaneka cell with 26.7% efficiency are 400 μm, 300 μm, and 165 μm thick, respectively. 2,4-6 Ray-optics based conventional cell designs not only portend a limiting efficiency below 29.43% but also involve considerable cost of the cell due to the requirement of high quality, thick c-Si.
An alternative to thick, inflexible, silicon solar cells is thin-film photovoltaics (TFPV). Typically, the light-absorbing layers of TFPV based cells are less than 10 μm thick and consist of direct bandgap materials such as cadmium telluride (CdTe), copper indium gallium diselenide (CIGS), gallium arsenide (GaAs), or perovskites. Among these technologies, GaAs cells are most efficient and have reached a conversion efficiency of 29.1% on a single-junction (world-record held by Alta Devices). 7 However, GaAs is very expensive due to the scarcity of Ga and the toxicity of As is undesirable. To date, the most efficient CIGS and CdTe cells have 22.9% (by Solar Frontier) 12 and 21% (by First Solar) 7 conversion efficiencies, respectively. Not only are the best CdTe and CIGS cells less efficient than the best silicon solar cells, but also the toxicity of Cd, scarcity and environmental concerns associated with Te, scarcity of Ga, slow and noneconomical production process of CIGS pose major challenges to the respective TFPV technologies. [13][14][15] Among the emergent photovoltaic technologies, perovskite or organo-lead halide cells have shown rapid efficiency growth in recent years. [16][17][18] Recently, 25.2% power conversion efficiency has been achieved in a thin-film perovskite cell. 18 The Shockley-Queisser limit for single-junction perovskite solar cells is approximately 31%, slightly less than that of silicon cells. Multi-junction cell architectures such as perovskite-silicon tandem have the potential to achieve higher efficiency. At present, Oxford PV holds the world-record efficiency of 28% for a perovskitesilicon tandem cell. 18 However, the lack of long-term stability and environmental safety concerns associated with lead (Pb) may outweigh the advantages of perovskite cells. This review focuses on combining the efficient light-trapping capability of photonic crystals (PhC) 19,20 with the non-toxicity, abundance, and well-developed fabrication techniques of Si to develop a cell technology that encompasses the advantages of the existing TFPV and paves the way toward and beyond 30% power conversion efficiency, a far-reaching goal of the photovoltaic industry.
PhC front-side filters have been used to improve performances of thermo-photovoltaic (TPV) cells [21][22][23][24][25][26][27] and photodetectors. 28,29 However, the photodiode materials in such applications usually have direct bandgaps. In contrast, our review focuses on unprecedented solar absorption in thin flexible, indirect bandgap silicon-PhCs, rivaling the absorption of much thicker conventional silicon solar cells and other direct bandgap materials. Although PhCs have been used to improve solar absorption in perovskite cells, 30,31 these cells have considerably lower photo-currents in comparison with the silicon-PhCs. Moreover, stability issues associated with the perovskites render these designs less desirable for large-scale practical deployment.
The most challenging task is to design thin-silicon PhCs that absorb sunlight beyond the Lambertian limit. Lambertian lighttrapping does not consider the wave nature of light. The derivation of this limit involves a series of assumptions. The Lambertian model involves a hypothetical randomly rough top surface of the cell, exhibiting no specular reflection of the incident sunlight at any wavelength. In reality, no surface can satisfy this assumption. A second assumption is that rays within the cell are deflected according to the cos θ probability distribution function, where θ is the angle relative to the normal. In contrast, wave-interference based light-trapping in photonic crystals, obtained by solving Maxwell's equations, gives rise to intricate energy flow patterns 32,33 such as parallel-to-interface refraction (PIR) of light and vortex-like energy flow. In the case of PIR, the incoming light is refracted along directions almost parallel to the top surface of the cell in contradistinction to the cos θ distribution of Lambertian ray-optics. 32 Furthermore, wave-interference gives rise to vortex-like energy flow patterns and slow-light modes inside a photonic crystal that cannot be described by ray-optics. 33 These interference effects lead to very long photon dwell-time in the active layer of the cell and appear as sharp resonant peaks in the absorption spectra of the cell. In contrast to a conventional, Lambertian cell that weakly absorbs sunlight in the 800-1100 nm wavelength range, a much thinner silicon PhC with wave-interference based resonances can absorb significantly more solar energy in this wavelength range. The strategy toward achieving above 30% power conversion efficiency in a silicon solar cell involves (i) light-trapping optimization in thinsilicon PhCs to achieve solar energy absorption exceeding the Lambertian limit, (ii) use of these optimized thin-silicon PhCs (much thinner than the diffusion lengths of the charge carriers) in order to collect more optically generated electron-hole pairs before they recombine, and (iii) optimization of the contact geometries and doping profiles of the cell to reduce the charge carrier recombination further.
We begin with a brief survey of the ray-optics-based Lambertian limit and high-efficiency conventional solar cells in Sec. II. We critically review the underlying assumptions and statistical arguments that lead to the 4n 2 optical path-length enhancement where n is the real part of the frequency-dependent refractive index of the light-absorbing material. Richter et al. 9 considered an ideal Lambertian cell free from both defect-mediated Shockley-Read-Hall (SRH) and surface recombinations. We revisit the Lambertian cell with more realistic finite SRH lifetime and surface recombination velocities (SRVs). In Sec. III, we review the progress made toward light-trapping and solar energy harvesting in photonic crystals. We compare the light-trapping performances of a wide variety of PhC architectures with unit cells consisting of nano-wires, [34][35][36][37][38][39][40][41] conical-pores, 33,42 inverted pyramids, [43][44][45][46] and parabolic-pores. 48,49 From this comparison, we discuss why the 10-15 μm-thick inverted pyramid PhC designs provide the best experimentally feasible route to solar energy absorption well beyond the Lambertian limit and silicon solar cells with conversion efficiency in the 28%-31% range.
Wave-interference-based light-trapping leads to very long dwell-times for long-wavelength photons inside the PhC thin-film and increases the probability that trapped photons get absorbed despite the indirect bandgap of silicon. We also consider electronic bandgap narrowing (BGN) 50 and phonon-assisted sub-bandgap energy absorption in c-Si. 51 46,47 In Sec. V, we consider the electronic performance of various cell designs and discuss carrier transport optimization strategies to maximize the collection of optically generated charge carriers in thin-film PhC cells. In contrast to the 110 μm optimum thickness of the hypothetical Lambertian cell, the optimum thickness of the silicon PhC cell with Auger recombination 11 and experimentally achievable carrier lifetime falls between 10 and 15 μm. This suggests that thin-film silicon photovoltaics has the potential to leap ahead of competing technologies and surpass the efficiency of any single-junction solar cell made of a single active material.

II. LAMBERTIAN LIGHT-TRAPPING AND EFFICIENCY LIMITS OF CONVENTIONAL SOLAR CELLS
The light-trapping capability of conventional solar cells typically falls below the ray-optics-based Lambertian limit. Ray-optics computations describe light absorption in conventional cells because the relevant feature sizes of these cells are very large as compared to the 300-1200 nm wavelength scales. For such large light-trapping geometries, ray-optics and statistical arguments suggest a putative upper bound of solar absorption. In a more realistic wave-optics picture, this is in fact only a benchmark rather than a true upper bound. To understand this Lambertian benchmark, we consider a slab of thickness L and refractive index n, with a randomly rough surface at the top and a perfectly reflecting mirror (or a perfect conductor) at the bottom (shown in Fig. 1). As the light rays enter the slab from air, they bounce back and forth multiple times between the top and bottom surfaces before escaping the slab. These multiple reflections enhance the effective light-path (Ltrap) inside the slab. The derivation of Ltrap is based on three crucial assumptions: 1. The top surface does not cause any specular reflection to the incoming light rays. 2. If a ray inside the slab is incident on the rough surface at an angle less than the critical angle θc (where sin θc = 1 n ) with respect to the surface normal, the surface allows perfect escape of the ray and causes total internal reflection otherwise. Thus, FIG. 1. Geometry used to model statistical ray-trapping in a slab with thickness L and refractive index n, placed on a perfect reflector. The randomly rough top surface is assumed to perfectly transmit all incoming rays into the slab and assumed to provide perfect total internal reflection for rays within the slab incident at an angle greater than the critical angle. the rays incident within a cone of angle θc with respect to the surface normal are able to escape the slab. 3. The top surface randomly redirects both the externally incident rays and total internally reflected rays within the slab at an angle θ with respect to the surface normal, according to a distribution function f (θ). According to assumptions 2 and 3, the probability p that the light escapes the slab is given by For a Lambertian surface, f (θ) = 1 π cos θ, 0 ≤ θ ≤ π 2 . 55 With this Lambertian distribution of the light rays, Eq. (1) yields p = 1 / n 2 . We define q to be the number of times a ray traverses the distance 2L and G(q) to be the probability that the ray escapes after the qth traversal. Thus, G(1) = p and G(q) = p(1 − p) q−1 [since the ray does not escape during the first (q − 1) traversal and escapes after the last traversal]. The average number of traversal can be calculated as ⟨q⟩ = q=1 ∑ ∞ qG(q) = n 2 . When a ray is reflected back to the slab at an angle θ, the path-length of the ray within the slab in each traversal is given by 2L /cos θ. Thus, the average path-length of the rays in a single traversal is given by For a Lambertian surface, the above integral yields Lavg = 4L. Thus, Ltrap can be evaluated from the average path-length in ⟨q⟩ traversals as Ltrap = ⟨q⟩ Lavg = 4n 2 L. Alternatively, we can say that the rate of escape of light trapped in the cell is v L trap , where v is the speed of light in the medium. In the literature, this statistical ray-trapping result is widely known as the 4n 2 limit or the Lambertian limit. The absorption coefficient (A Lambertian ) of a slab of thickness L and complex refractive index (nr − iκ) can then be calculated as a ratio of the rate of absorption to the combined rate of absorption and escape, 10,33 Here, α is the wavelength (λ) dependent intrinsic inverse absorption length of the slab material, given by α = 4πκ / λ . Richter et al. 9 modeled maximum light absorption of a silicon solar cell using Eq. (3) and calculated the 29.43% efficiency limit and optimum cell-thickness of 110 μm at 25 ○ C for a single-junction cell using a revised model of Auger recombination. 11 However, this estimation ignores all SRH recombination and surface recombination. Moreover, it considers Auger recombination for the case of undoped silicon. In order to compare with the performance of the hypothetical Lambertian cell with a realistic cell, we present a revised version of the Lambertian efficiency limit in Sec. V by taking into account realistic SRH lifetime (τSRH), surface recombination velocity (SRV), and an appropriate doping profile.
Conventional light-trapping designs of the high-efficiency solar cells rely on ray-optics. Accordingly, the 29.43% limit applies well to cells that satisfy the underlying assumptions. For example, the loss PERSPECTIVE scitation.org/journal/app analysis of the 26.7% Kaneka cell shows that with perfect Lambertian light absorption, no extrinsic recombination (SRH and surface recombination), and no resistive losses, it would reach a conversion efficiency of 29.1%. This is precisely the efficiency limit of a hypothetical Lambertian cell with a p-type substrate having the same thickness (165 μm) and resistivity (3Ω cm) as the Kaneka cell. 6 Two of the commonly employed ray-trapping architectures in high-efficiency silicon solar cells are pyramids and inverted pyramids. Pyramidal textures on c-Si can be fabricated very easily using wet-etching of silicon with KOH. However, in conventional raytrapping, these pyramids have length scales much larger than the wavelength of light. 1,2,56-61 For such cells, the pyramid sizes and their arrangements can either be uniform 2 or random. 58,62 Light absorption in solar cells, textured with regular or random arrays of upright pyramids with 10 μm or larger base-lengths, falls below the Lambertian limit even when the silicon thickness approaches 400 μm. 56,57 Ray-trapping in 180 μm-thick solar cell textured with random arrays of inverted and upright pyramids with ∼15 μm base-length yields a photo-current density of 36 mA/cm 2 . Light absorption in such cells can be modeled accurately using ray-optics based computations. [63][64][65] However, these thicker cells are inflexible. The high non-radiative bulk-recombination of charge carriers is a major roadblock toward achieving higher efficiency in such cells.
Another popular ray-trapping architecture for photovoltaic applications is a random arrangement of silicon nano-pillars 66 with random cross sections, also known as black silicon. [67][68][69] So far, the power conversion efficiencies of some of the best black silicon cells are 22.1% and 23.5%, 70,71 still well below the putative efficiency limit of 29.43%.

III. WAVE-INTERFERENCE BASED LIGHT-TRAPPING IN PHOTONIC CRYSTALS
The lattice constants of PhCs are comparable to the wavelength of light. In thin-film PhC solar cells, the overall thickness of the structure is only a small number of wavelengths. In this situation, modeling of light-waves by rays would result in an inaccurate description of energy propagation. Instead, an accurate numerical solution of Maxwell's equations is needed in order to accurately capture the light-trapping effects. Light-trapping in PhCs was originally associated with the complete localization of light within a structure through suppression of electromagnetic density of states in the form of a photonic bandgap. 19,72 In contrast, efficient absorption of broadband light, from an external source, arises from the ability to enhance the electromagnetic density of states in a specific frequency regime of PhCs. 32 The design and fabrication of thinfilm PhC nanostructures for photovoltaic applications have been very actively pursued using nano-wire, 34-41,73-81 nano-cone, 33,42,75 and nano-pyramid architectures. [43][44][45]82 Other complex-shaped unit cells such as roof-mosaic, rose, and zigzag with lattice-periodicity in 920-1000 nm range has also been considered for 20 μm-thick c-Si. 82 Light-trapping approaches, combining PhC and plasmonic resonances, have also been proposed. [83][84][85] However, parasitic absorption of light in the metal particles and recombination losses at the unpassivated metal-semiconductor interfaces reduce the power conversion efficiency of such cells.
The amount of light absorption in a solar cell is quantified by the maximum achievable photo-current density (MAPD). Under AM1.5G illumination, the MAPD of a cell is given by Here, I(λ) is the intensity of the AM1.5G spectrum; A(λ) is the wavelength dependent absorption of the particular structure; λmax is the maximum wavelength of light that the cell can absorb; and e, h, and c are the electronic charge, Planck's constant, and speed of light, respectively. It is assumed that each absorbed photon creates a single electron-hole pair. In the absence of bulk and surface recombination losses, JMAPD is equivalent to the short-circuit current (JSC) of the solar cell. In the absence of BGN, λmax = 1100 nm for silicon cells. However, with the inclusion of BGN and phonon-assisted absorption in the Urbach tail, [51][52][53][54] λmax is chosen to be 1200 nm.
In order to surpass the 30% efficiency barrier, it is necessary to optimize the light-trapping in a thin-film PhC such that its solar absorption exceeds the Lambertian limit and rivals that of a conventional cell 20 times thicker. A key strategy to achieve this goal is to harness the capability of a thin-film PhC to absorb substantial sunlight beyond 800 nm wavelength, a regime where conventional cells are very weak absorbers of solar energy. Early studies focused on ultra-thin silicon structures with an equivalent bulk thickness on the order of 1 μm. Some of these studies considered structures that are not easily fabricated but through which design principles and physical phenomena could easily be established. Using the finite difference time domain (FDTD) 86 method to solve Maxwell's equations, 87 it was shown that modulated silicon nanowires, arranged in a square lattice and embedded in a SiO 2 background, yield a MAPD of 27.7 mA/cm 2 , over the 350-1000 nm range, using only 1 μm equivalent c-Si thickness. 40,41 While this PhC is able to absorb 63.7% of the incident solar energy using just 1 μm equivalent siliconthickness, the MAPD still falls below the Lambertian absorption limit. These nanowires have a significant surface area. Using a modest surface recombination velocity of 100 cm/s at the Si-SiO 2 interfaces surrounding each nanowire and a relatively poor recombination velocity of 10 5 cm/s at the bottom metallic contact, a power conversion efficiency of 15% was predicted. 41 While this is far below high-efficiency solar cells, it is noteworthy that with only 1 μm of silicon and suitable geometrical structure, wave-interference effects such as PIR are manifest and moderate efficiency is nevertheless possible.
Although it is relatively easy to surpass the Lambertian limit over a narrow range of frequencies and angles of incidence, it is a challenging task when the entire solar spectrum and a wider angular range are considered. Slanted conical-pore photonic crystals achieve this difficult goal using multiple resonances in the 800-1100 nm wavelength range. These exhibit PIR and vortexlike flow inside the thin-silicon PhC. 33 For 1 μm equivalent bulkthickness of c-Si, the optimal base-diameter of the slanted cone is 1000 nm and lattice constant of the PhC is 850 nm. This PhC architecture, backed by an Ag back-reflector, yields a MAPD of 35.5 mA/cm 2 over the 300-1100 nm wavelength range, exceeding the Lambertian limit of 32.86 mA/cm 2 for 1 μm-thick c-Si. This study also shows that the response of the slanted conical-pore PhC is extremely robust with respect to the angle of incidence and the PERSPECTIVE scitation.org/journal/app absorption does not exhibit any substantial degradation at least up to an off-normal incidence angle of 50 ○ for all polarizations. This extreme robustness of slanted-cone PhC is due to the PIR modes that exhibit similar robustness for a wide range of angles of incidence. 32 x-y symmetry-breaking slanted conical-pore PhC enables better light-trapping than its vertical counterpart. This was illustrated in Ref. 33 by comparing the performances of the slanted and vertical conical-pore PhCs placed on semi-infinite glass substrates. These ultra-thin silicon structures demonstrate the possibility of solar light-trapping and absorption beyond the Lambertian limit over the entire 300-1100 nm wavelength range. However, they are difficult to fabricate on a large scale and the required pore shapes may give rise to unwanted surface recombination centers of photogenerated charge carriers. Nevertheless, assuming a SRV of 100 cm/s throughout the top Si-SiO 2 interfaces and a bottom contact SRV of 10 5 cm/s, a projected efficiency of 17.5% was predicted 42 for a silicon film with only 1 μm equivalent bulk thickness. This efficiency improves to 22.5% 42 if the bottom contact SRV is reduced to 100 cm/s. In these calculations, a detailed treatment of Auger and Shockley-Read-Hall carrier recombination was not undertaken. Instead, it was assumed that the combined effect of these recombination processes simply led to an overall carrier diffusion length of about 10 μm. Similar considerations apply to other direct bandgap semiconductors such as gallium arsenide. Eyderman and John 88 illustrated absorption of almost 90% of the incident sunlight over the 300-865 nm wavelength range in a slanted-conical pore PhC with 200 nm equivalent bulk-thickness of GaAs. The slanted conicalpores have a base-diameter of 600 nm, pore-depth of 290 nm, and lattice constant of 550 nm. In contrast to silicon, GaAs is a direct bandgap material and susceptible to significant radiative recombination. The photons emitted due to radiative recombination can be reabsorbed by the GaAs cell. This is known as photon-recycling. In the absence of photon-recycling and for 1000 cm/s SRV, the slanted-conical pore GaAs cell yields 28.3% power conversion efficiency with a MAPD of 27.6 mA/cm 2 . With complete photonrecycling and the same SRV, the conversion efficiency increases to 29%. Surface passivation for GaAs is more difficult than for silicon, and it may be practically unfeasible to achieve SRV below 1000 cm/s. If the SRV is hypothetically reduced to 10 cm/s, the conversion efficiency of the ultra-thin cell would reach 30.6%. Although slanted-conical pore PhC's exhibit excellent light-trapping capability and beyond-Lambertian limit absorption, fabrication of such structures is difficult and may require advanced nano-imprinting and deposition techniques. Due to the direct bandgap of GaAs, photonic crystal light-trapping serves primarily to reduce the thickness of the cell by about a factor of 10 from the current worldrecord holding cell from Alta Devices. 7 Efficiencies beyond the current 29.1% would require improvements to state-of-the-art surface passivation.

PERSPECTIVE scitation.org/journal/app
The requirements that thin-film silicon PhCs absorb light beyond the Lambertian limit and at the same time are amenable to existing fabrication techniques lead us to parabolic-pore and inverted micro-pyramid architectures. Using reactive ion etching (RIE), Kuang et al. fabricated 10 μm-thick c-Si parabolic pore PhC with 1200 nm lattice constant and 60 nm anti-reflection coating (ARC) of SiO 2 . 48 In this PhC architecture, each pore has a depth to lattice constant ratio of 0.65. FDTD computations exhibit excellent agreement with the measured absorption of this PhC and reveal a MAPD of 39.1 mA/cm 2 , slightly below the Lambertian limit of 39.65 mA/cm 2 in a 10 μm c-Si slab. High resolution FDTD simulations also reveal a high density of resonances due to PIR in the near-infrared spectrum. This structure combines a graded-index anti-reflection property and strong absorption of sunlight through wave-interference based light-trapping. An experimentally feasible way to enhance the light absorption is by simultaneously optimizing the pore-height (h) and lattice constant (a) of the PhC. 49 This (h/a) ratio is crucial to optimizing light-trapping. For a more graded-index and anti-reflection, a larger (h/a) is desirable. However, deeper pores imply less light-absorbing material. Figure 2(a) shows a schematic of the parabolic-pore PhC based solar cell design proposed in Ref. 49. This passivated emitter rear contact (PERC) cell has a conformal n-type region of width temitter with a doping density N d and a p-type base region with a doping density Na. The combination of the perfect electric conductor (PEC) layer of thickness tPEC = 100 nm and SiO 2 buffer layer of thickness t buffer = 75 nm underneath the cell acts as a back-reflector of the cell and improves the optical performance of the cell. If the PEC layer is replaced by a 100 nm thick Ag layer, the same MAPD is obtained, provided the buffer layer exists between the silicon and the Ag layer. 49 This buffer layer also acts as rear passivation to the cell. The bottom contact touches the active silicon region through highly p-doped (p + ) back surface field (BSF) regions. Similarly, the front SiO 2 anti-reflection coating (ARC) of thickness tARC provides front passivation. The front electrodes of the cell, made of either indium tin oxide (ITO) or metal, make contact to c-Si through highly n-doped (n + ) regions, diffused at a regular spacing determined by the emitter-pitch of the cell.
The optimization map of Fig. 3 (assuming t buffer = 75 nm and no ARC), obtained from 3D FDTD computations using the Electromagnetic Template Library (EMTL), 89 shows that the optimum design parameters for the 10 μm-thick c-Si parabolic-pore PhC are a = 1000 nm and (h/a) = 1.7. This provides a MAPD of 40.57 mA/cm 2 , already more than the Lambertian limit of 39.65 mA/cm 2 . With a 50 nm thick non-conformal ARC layer of SiO 2 , this MAPD improves to 41.09 mA/cm 2 . As in the case of conical-pore PhC, 33 a further improvement in the light absorption of the parabolic-pore PhC can be achieved by tilting the pores slightly and breaking the xy symmetry of the crystal. An optimum tilt angle of θ = 10 ○ enhances the light absorption in the 750-1000 nm wavelength range of the solar spectrum (see Fig. 4) and yields MAPD of 41.72 mA/cm 2 . 49 The solid blue circle in Fig. 4 corresponds to the 97.4% absorption peak at λ = 940 nm for the slanted parabolic-pore PhC. In comparison with this, the vertical parabolic-pore PhC absorbs 86.7% light at this wavelength.
The underlying mechanism for the better light-trapping is illustrated through comparison of the in-plane Poynting vector flows of the vertical and tilted pore structures. For the x-y symmetric PhC, almost all the incident energy flows along the z-direction. When the x-y symmetry is broken, the Poynting vector exhibits considerable parallel-to-interface energy flow and vortex-like patterns near the intensity hotspots. 49 The resulting long dwell time of photons within the active region facilitates carrier generation even when the intrinsic absorption of silicon is weak. Parallel-to-interface and vortex-like flow of light is the result of multiple coherent scattering and wave-interference. They cannot be captured in a ray-optics approximation. The Lambertian distribution predicts negligible probability for ray propagation nearly parallel to the air-silicon interface. In contrast, an accurate solution of Maxwell's equations shows that parallel-to-interface and vortex-like energy flows are very prominent in well-designed photonic crystals beyond 700 nm wavelength. These and other slow-light modes enable thin-silicon PhC structures to absorb sunlight beyond the Lambertian limit. 1D charge carrier transport calculation reveals that a 10 μm-thick slanted parabolic-pore PhC solar cell is capable of achieving 28%-29% power conversion efficiency when including absorption only in the 300-1100 nm range. 49 However, this efficiency does not include practical effects such as electronic BGN 50 and phonon-assisted sub-bandgap absorption associated with the Urbach tail. [51][52][53][54] While parabolic-pore structures are amenable to microfabrication, the reactive-ion etching method may cause substantial damage to the silicon crystal structure below the top surface. The resulting electronic defects may lead to considerable non-radiative carrier recombination in the bulk. The slightly overlapping pores in the structure (optimized for light-trapping) lead to intricate teepeelike protrusions that may likewise act as surface recombination centers. Both of these effects, without any further treatment, may lead to short carrier lifetime and reduce the solar cell efficiency. It is, therefore, vital to consider architectures in which both light-trapping and electronic properties can be optimized.

IV. MICRO-PYRAMID PHOTONIC CRYSTAL DESIGN OPTIMIZATION
A more likely candidate for high efficiency silicon solar cells consists of inverted-micropyramid PhCs. These can be fabricated with a high accuracy through masking and wet-etching of c-Si. Due to the anisotropic etching property toward c-Si, KOH readily etches and exposes the (111) and related surfaces of silicon, leading to formation of inverted-pyramids with a side-wall angle of 54.7 ○ , the PERSPECTIVE scitation.org/journal/app angle between (111) and the initial (100) planes of c-Si. A detailed FDTD based light-trapping optimization study 44 of silicon invertedmicropyramid PhCs showed that a 10 μm-thick structure with a lattice constant of 2500 nm and 200 nm SiO 2 ARC enables MAPD of 42.5 mA/cm 2 , significantly surpassing the Lambertian limit. This architecture differs from the classical inverted-pyramid PERL cells, used to achieve an early world-record efficiency, 2 in two important ways. The first is the choice of the pyramid base length and corresponding pitch of the square lattice array. The second is the thin and flexible nature of the PhC architecture. Unlike the classic PERL cell pyramids with base lengths on the order of 10 μm, the photonic crystal micro-pyramids focus on base lengths and lattice constant in the range of 1-3 μm, much closer to the wavelength of sunlight. This reduction in length scales leads to dramatically different wave-interference based light-trapping effects that are absent in the classic structures that rely on ray-optics based light-trapping. The result is that a 10 μm-thick silicon micro-pyramid PhC can absorb as much light as a 400 μm-thick classic inverted-pyramid PERL cell. This dramatic reduction in thickness, while not sacrificing solar-absorption, has the major consequence that bulk, nonradiative charge carrier recombination (including Auger recombination) is substantially reduced. The inverted micro-pyramid PhC has the advantage over previously discussed PhC architectures in that the exposed (111) face of silicon is amenable to high-quality surface passivation. In other words, the micro-pyramid PhC enables both state-of-the-art electronic quality and beyond-Lambertian light-trapping. The optical and electronic modeling of inverted micro-pyramid PhCs, made from 3 μm to 20 μm-thick c-Si, has been reconsidered in recent works. 46,47 Two types of inverted micro-pyramid PhC solar cell architectures, PERC (similar to the parabolic-pore PhC solar cell shown in Fig. 2) and interdigitated back-contact (IBC) cells, are shown in Figs. 6(a) and 6(c), respectively. The PERC architecture features 10-20 μm-wide metal contacts at the top of the cell. These metal contacts block a part of the incoming sunlight, leading to "shadowing loss" in the cell. Experimental measurements 90 have shown that a front-contact geometry with 20 μm finger-width and 800 μm finger-spacing leads to ∼1% shading loss of the photocurrent in the solar cell. The loss in the conversion efficiency of the cell is roughly the same as this percentage shading loss. Some recent design and experiments of "cloaked contacts" have shown that a major fraction of this loss can be overcome by redirecting the light around the contacts. 91-94 A more reliable way to eliminate the shading loss is to place all the contacts at the rear surface of the cell, leading to the IBC architecture. [95][96][97] The most recent worldrecord holding Kaneka cell 6 uses the IBC architecture. In addition to shading-loss, the PERC architecture also suffers from loss in power conversion efficiency due to sheet-resistance. The carriers, entering the emitter region of the cell, must travel a long lateral distance of several hundreds of micrometers between the front electrodes

PERSPECTIVE
scitation.org/journal/app through the thin emitter-sheet before they can be collected. The corresponding resistance contributes to a loss in the conversion efficiency. While it is tempting to reduce the sheet-resistance by increasing the emitter-thickness, doing so would also cause more Auger recombination of carriers. In order to balance the Auger recombination and sheet-resistance loss, an optimum emitter-design must be chosen for a PERC cell. On the other hand, when the emitter and base contacts are placed alternately in the IBC cells, they do not suffer substantial loss in efficiency due to emitter resistance. Overall, the IBC design provides better solar absorption, eliminates sheet resistance losses, and improves overall charge carrier collection, leading to the highest possible solar cell efficiencies. The ARC layers also act as front passivation for the cell. The electrode fingers make contact with the c-Si layer through highly doped regions with donor (n + ) and acceptor (p + ) impurities. The strong electric field between the highly doped, thin n + layer and ptype bulk keeps the minority carriers (holes) away from the emitter electrodes and helps in efficient collection of the photo-generated electrons. Similarly, the p + layer acts as the BSF region, leading to efficient collection of holes in the base electrodes. The Gaussian doping profiles of the n + and p + regions are characterized by Ni = N i0 exp(−z 2 /2σ 2 i ), where Ni 0 denotes the peak concentration (in cm −3 ) of the dopant species, σi is the Gaussian widths (in nm), and i = n and p for the n + and p + regions, respectively. For both cells, the bulk p-region is assumed to have a uniform acceptor doping concentration (NA) of 5 × 10 15 cm −3 . The top contacts of the PERC cell are assumed to be cloaked. The IBC cell has a p + front surface field (FSF) layer below the front-passivation. The p + and n + regions of the IBC cell that cover the base and emitter contacts have widths w pdop and w ndop , respectively, and are separated by a distance wpn. Parts of the back-reflectors protrude through the rear passivation into the silicon and form base and emitter contacts. The widths of the base and emitter contacts are denoted by wpcon and wncon, respectively.
In FDTD calculations, c-Si is modeled according to a modified Lorentz model over the 300-1100 nm wavelength range. 98 However, external quantum efficiency (EQE) measurements show that c-Si solar cells are capable of collecting carriers generated by light absorption up to 1200 nm. [99][100][101][102][103][104] This sub-bandgap absorption in c-Si originates from two mechanisms. The first one is carrier concentration dependent BGN in silicon. This is modeled using Schenk's model 50 and allows absorption of sunlight up to 1120 nm. 46 The absorption up to 1200 nm in c-Si, demonstrated by the EQE measurements, originates from the exponentially decaying Urbach optical absorption edge below the continuum band edge. 51,52 Static disorder in non-crystalline solids leads to an exponential band tail of localized states below the electronic band edge. However, in c-Si, a similar tail of phonon assisted optical absorption generates mobile electron-hole pairs. 53 where each term of the summation represents a Lorentz oscillator. The fitting parameters ϵ∞, ωpj, △ϵj, and γj (shown in Table I), used in the FDTD computation, are obtained using a freely available MATLAB program. 106 The experimental data are found to exhibit an Urbach slope of 8.6 meV over the 1160-1190 nm wavelength range. 46 Figure 7(a) shows the MAPD optimizations over the 300-1100 nm wavelength range for the inverted pyramid PhCs with different c-Si thicknesses. The front surfaces of the PhCs are coated with optimized dual-layer ARC's with n 1 = 1.4, t 1 = 45 nm, n 2 = 2.6, and t 2 = 100 nm. 46 We tabulate the details of the optimization results, along with the MAPD gain, due to sub-bandgap absorption in the 1100-1200 nm wavelength range in Table II. We also include the MAPD corresponding to the Lambertian limit for different values of c-Si thickness. This shows that for each cell-thickness over the 3-20 μm range, the MAPD of inverted micro-pyramid PhC surpasses the Lambertian limit by a significant margin. In Figs. 7(b) and 7(c), we compare the Lambertian absorption spectra (plotted in red) with those of the optimized inverted micro-pyramid PhCs (plotted in blue) for H = 15 μm and 5 μm. 47 The PhC absorption spectra exhibit significant light absorption through high densities of resonance peaks in the 950-1200 nm wavelength range. Figure 7 xz-plane of the 5 μm-thick, optimized PhC unit cell. This wavelength corresponds to a sharp resonance peak in the PhC absorption spectra of Fig. 7(d). The Poynting vector plot reveals parallel-to-interface and vortex-like energy-flow, indicative of wave-interference based light-trapping. It is worthwhile asking at this point whether all of the absorption predicted in the 1100-1200 nm range is in fact useful to the purpose of generating photo-current. Absorption that occurs in subgap, localized, electronic states (arising from static disorder) may only lead to rapid charge carrier recombination. However, subgap absorption that is assisted by simultaneous phonon absorption can elevate the excited electron into extended photo-current carrying levels. In order to address this issue, we provide in Table III a     wavelength cutoff. This is presented for the case of the 15 μm-thick cell for cutoff wavelengths ranging from 1100 to 1200 nm. The conditions leading to the projected power conversion efficiencies are described in the following section.

V. ELECTRONIC PERFORMANCE OF THIN-FILM PHOTONIC CRYSTAL SOLAR CELLS
In this section, we illustrate the electronic performances of various PhC solar cells through numerical solutions of the 1D and 2D Poisson's equation, coupled with semiconductor drift-diffusion equations. The numerical calculations are performed using Sentaurus. 107 We review 1D and 2D transport calculations for the inverted pyramid PhC PERC cell and 2D transport calculations for the IBC cell. These 1D and 2D charge carrier transport calculations have been shown to agree well with more demanding 3D calculations. 46 For 1D transport calculations, MAPDs from 3D FDTD computations are used to obtain an equivalent generation profile according to an algorithm and the efficiency is corrected for sheet resistance losses. 49 For 2D transport calculations, the actual generation profiles obtained from the 3D computations are integrated along one of the lattice directions. The resulting 2D profiles are then repeated over many PhC unit cells to cover the large scale contact geometry used in the Sentaurus computations. The 2D carrier generation profiles for the optimized inverted pyramid PhCs with H = 3 μm, 5 μm, 7 μm, and 10 μm, obtained by integrating the 3D generation profile along the y-direction over a unit cell, are shown in Fig. 8.
The charge carrier transport calculations of thin-silicon PhC solar cells employ various models for bulk and surface recombination. 46,47,49 The bulk-recombination of the carriers takes into account both Auger (implemented using the state-of-the-art Auger model in Ref. 11) and SRH recombination (using a SRH lifetime, τSRH). Due to significantly reduced bulk recombination enabled by optimized doping of the thin silicon layer, the diffusion length of the charge carriers is much larger than the PhC solar cell thickness. As a result, most of the photo-generated carriers are able to reach the surface of the cells. However, the potential to realize a high-efficiency cell depends on the ability of the cell to collect these carriers before they recombine at the interfaces. Surface passivation is crucial to realize high-efficiency thin-silicon PhC cells. State-of-the-art surface recombination velocities at the oxidesilicon interface typically fall in the range of 1-10 cm/s. 11,108,109 However, the SRV at the metal-silicon contact is much higher. Instead of direct contact to a metal, high efficiency solar cells use The lattice constants correspond to optimized PhCs. In 3D, thin wedges of silicon occur within the triangles, leading to carrier generation throughout the depicted 2D regions.

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scitation.org/journal/app carrier-selective heterojunction contacts. These include a stack of intrinsic and p/n-doped hydrogenated amorphous silicon (as employed in the Kaneka cell 6 ) or polysilicon on oxide (as employed in the POLO-IBC cell 4 ). The SRV at the interface of c-Si and a stack of intrinsic and doped amorphous silicon can be as low as 2-5 cm/s. 110,111 For POLO-IBC cells, the contact SRV ranges between 2 cm/s and 20 cm/s. 5 In 1D transport calculations, the geometric details of the passivation layers and contacts of the actual 3D cell are subsumed into the equivalent front and rear surface recombination velocities, SRV front and SRVrear. 46,49 These equivalent SRVs appear as boundary conditions in the 1D transport model. For 2D and 3D computations, a more microscopic model is used for the surface recombination at the insulator-Si interface using SRH statistics for the recombination rate, 112 where Sj 0 = v th,j σjD interface with j = n, p are microscopic surface recombination velocities (v th,j is the thermal velocity of the electrons/holes, σj is the electron/hole capture cross section, and D interface is the interface trap density at the oxide-semiconductor interface), ns and ps are the electron and hole concentration at the Si surface, and Here, T is the temperature (in K) and Eg(T) denotes the bandgap of Si. Ne and N h are defined in terms of the electron/hole effective mass m * e /m * h and Planck's constant h as with j = e and h for electrons and holes, respectively. For electrons, v th = √ 3KT m * e = 1.12 × 10 7 cm/s for m * e = 1.08me and T = 298 K. The thermal velocity of holes is slightly lower due to higher effective mass (∼ 1.5m * e ). D interface is set to 3 × 10 9 cm −2 according to the measured value of the near-midgap trap density at the Si-insulator interface in Ref. 113. σp is taken to be 6 × 10 −17 cm 2 for these traps from the measured data on capture cross sections (Fig. 6 in Ref. 114). This figure also shows that the measured value of σn varies over a large range. The choice of σn = 6 × 10 −16 cm 2 results in Sn 0 that closely approximates the effective SRV of state-of-the-art measurements. 11 Accordingly, the choice of Sn 0 ≈ 20.16 cm/s and Sp 0 ≈ 1.7 cm/s is made for the 2D transport calculations 46,47 discussed in this review.  The overall effect is a slight lowering of the cell efficiency. In optimized photonic crystal solar cells, wave-interference based light-trapping provides a more significant increase in photo current that more than offsets the reduction in voltage. The result is a slight increase in efficiency. 46,47 In the 2D transport calculation of Fig. 9, BGN is included using Schenk's model 50 and an accurate description is made of the c-Si optical absorption in the 1100-1200 nm wavelength range (given in Table I). State-of-the-art lifetime measurements 11 show that a good quality c-Si wafer has τSRH close to 10 ms. Using these parameters, along with an emitter-pitch of 500 μm, a 30.15% conversion efficiency is predicted for the 10 μm-thick, inverted micro-pyramid PhC solar cell with cloaked upper contacts. The corresponding J-V  9 and current world-record efficiency of 26.7% (obtained using an ∼16 times thicker cell) 6 by a significant margin. We now describe the electronic performance of the inverted micro-pyramid PhC cell with an IBC architecture. This architecture is preferred over the PERC architecture due to the absence of shading-loss and sheet-resistance loss. The details of the design parameters of an PhC IBC cell, with a uniform bulk acceptor doping of 5 × 10 15 and 1 μm separation between n + and p + regions at the back of the cell, are shown in Table IV. The parameters such as wpcon, wncon, Np 0 , σp, Nn 0 , and σn are optimized to yield maximum conversion efficiency. 47 The same modeling parameters are used for BGN, sub-bandgap absorption, and surface recombination at the insulator-silicon interface as in the case of the PERC cell.
Unlike the PhC PERC cells, the thickness optimization of the PhC IBC cell is presented for a range of values of τSRH. 47 The reason for this is that τSRH is determined by the bulk defects in the c-Si wafer and 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 a 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 slightly larger. In Fig. 10, we use τSRH = 0.1 ms, 0.5 ms, 1 ms, and 10 ms and show the optimum cell-thickness for each case (other simulation parameters appear in Table IV). Figures 10(b)-10(d) show the variation of VOC, JSC, and FF as a function of the cell-thickness. As the cell-thickness is increased from 3 μm to 20 μm, the FF of the IBC cell drops by 4% (additive) for τSRH = 0.1 ms. As τSRH increases, the drop in the FF becomes smaller for the same range of cell-thickness variation. For τSRH = 10 ms, the FF becomes almost independent of cell-thickness.

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scitation.org/journal/app 1% (additive) in comparison with the PhC PERC cell and leads to a conversion efficiency more than the world-record efficiency of any single-material cell. This also suggests the possibility of a poly-silicon solar cell of 7 μm thickness and τSRH ∼ 0.1 ms, achieving a record efficiency of 27%. Such a cell may be amenable to large-scale, low-cost fabrication through deposition methods. The 29.43% Lambertian efficiency limit of a silicon solar cell is calculated assuming hypothetical material properties such as no SRH and surface recombination and undoped bulk silicon. 9 To compare the performance of the PhC IBC cell to that of a realistic cell with Lambertian light-trapping, it is necessary to include the same doping profiles, surface recombination, SRH lifetime, and IBC contact geometry as used in the PhC IBC cell. Figure 11 shows the performance parameters of more realistic Lambertian IBC cells of varying thickness, designed according to the parameters of Table IV and τSRH = 10 ms. The optimum thickness of the Lambertian cell (with realistic material parameters) is found to be 90 μm with a maximum conversion efficiency of 28.37% in comparison with the optimum 110 μm-thick hypothetical Lambertian cell with 29.43% conversion efficiency. In practice, the light-trapping in the conventional cells falls below the Lambertian limit. This results in an increased optimum thickness of the conventional cells with practical ray-trapping.
In thin-silicon PhC cells with optimized doping configurations, the photo-generated charge carriers travel a small distance in comparison with their diffusion lengths before they are collected at the electrodes. Such cells benefit from low bulk-recombination loss. However, surface recombination remains a very important factor. In Fig. 12, it is shown that the optimization of the FSF and BSF regions plays a paramount role in keeping the minority carriers away  47 Improved Auger, 11 Contact SRVs doping profiles τSRH = 10 ms 10 cm/s in Table IV Inverted pyramid Improved Auger, 11 Contact SRVs PhC IBC, 2D τSRH = 10 ms 10 cm/s transport 47 Table IV) Inverted pyramid Improved Auger, 11 Contact SRVs PhC IBC, 2D τSRH = 10 ms 100 cm/s transport 47 Figure 12(a) shows that the power conversion efficiency of the IBC cell with optimized FSF and BSF (i.e., Np 0 = 4 × 10 18 cm −3 and σp = 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). 47 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 (Np 0 = 1 × 10 17 cm −3 and σp = 100 nm in this particular example). When the contact SRVs are extremely high (∼10 6 cm/s), the IBC cell with optimum FSF/BSF doping retains ∼20% power conversion efficiency. This is in sharp contrast to the cell with inadequate FSF/BSF where the conversion efficiency drops to ∼5%. A similar role of BSF is apparent 46 in the case of inverted micro-pyramid PhC PERC cells.
We compare the performance parameters of the thin-silicon inverted micro-pyramid PhC cell with those of the much thicker hypothetical Lambertian cell and the current world-record holding Kaneka cell. The efficiency limit analysis of the hypothetical Lambertian cell assumes no bulk SRH and surface recombination. In addition, the quoted 29.43% efficiency limit corresponds to an undoped bulk silicon. 9 As discussed earlier, a more realistic cell with perfect Lambertian light-trapping would have a lower efficiency. The optimum thickness of a Lambertian cell with a similar doping, SRH recombination, and SRV as the PhC cells described in this article is slightly reduced to 90 μm and has 28.37% power-conversion efficiency. Table V shows that the 15 μm-thick IBC cell, with experimentally attainable design parameters, surpasses the efficiency of a Lambertian cell with a practical carrier transport model by 2.7% (additive). Both inverted micro-pyramid PERC and IBC cells exceed the conversion efficiency of the world-record holding Kaneka cell by ∼3.4% (additive) and 4.37% (additive), respectively. In contrast to the Kaneka cell that uses n-type bulk, the recently developed 300 μm-thick POLO-IBC cell 4,5 and the PhC cells reviewed in this article use p-type bulk.
We note finally the consequences of light absorption in the solar cell that might not contribute to photo-current at all. In the optimized 15 μm-thick, inverted micropyramid cell, wave-interferencebased light-trapping, in the 1100-1200 nm wavelength range, is predicted to contribute 1.36 mA/cm 2 toward the overall MAPD. Table III summarizes the consequence of removing some or all of this absorption from the carrier generation rate. For all choices of the absorption cutoff wavelength, from 1100 to 1200 nm, the resulting power conversion efficiency remains above 30%.

VI. CONCLUSION
Over the past three decades, light-trapping designs of solar cells have been dominated by ray-optics concepts. Significant progress has been made toward improving wafer quality and contact architectures and reducing surface recombination. However, the lighttrapping mechanisms for high-efficiency silicon solar cells have largely ignored the wave nature of light. Despite the progress in electronic properties, the power conversion efficiency has almost reached a saturation level as evidenced by the 1.7% (additive) efficiency increase in the single-junction silicon solar cells over the past two decades. This review highlights the importance of wave-interference based light-trapping to overcome previously reported barriers to solar absorption and cell efficiency. This waveinterference-based solar absorption is optimized in certain photonic crystal architectures with features on the scale of near-infrared light. In particular, using 10-15 μm-thick inverted micropyramid photonic crystals, it is possible to substantially surpass the Lambertian limit for solar absorption over the 300-1200 nm wavelength range. This opens an unexplored avenue for the realization of singlejunction silicon solar cells with a new limiting efficiency of 31%, which is much higher than previous estimates. Remarkably, such high-efficiency silicon solar cells may also be thin and flexible, making them suitable for integration into buildings and other powerconsuming devices. Silicon, despite being an indirect bandgap material, is projected to surpass the efficiencies of other direct bandgap materials. The numerically demonstrated power conversion efficiency in the 15 μm-thick PhC-IBC cell is significantly higher than the best thick-silicon cells, such as the 26.7% Kaneka cell and 26.1% POLO-IBC cell, and even higher than the best direct bandgap GaAs cell with 29.1% efficiency.
The efficiency estimate of 31% for the 15 μm-thick silicon PhC solar cell relies on a combination of different factors. These include (i) slightly over 1 mA/cm 2 of useful photocurrent harvested from 1100 to 1200 nm wavelength band, (ii) a carrier lifetime of 10 ms, and (iii) surface recombination velocities on the order of 10 cm/s. However, there is considerable robustness to the prediction of beyond 30% power conversion efficiency. Even if part or all of the sunlight from the 1100-1200 nm band is removed, the projected efficiency remains in the 30%-31% range (see Table III). The same holds true if the carrier lifetime drops to 1 ms [see Fig. 10(a)]. However, as the carrier lifetime is reduced, the optimum thickness of the solar cell is correspondingly reduced. For a more dramatic drop in carrier lifetime to 0.1 ms, the optimum cell-thickness drops to 7 μm and the resulting structure is projected to yield a power conversion efficiency of 27%. Finally, an increase in surface recombination velocity to 100 cm/s still yields above 30% efficiency [see Fig. 12(a)], provided that the silicon doping profiles are chosen appropriately.
Remarkably, the potential efficiency of thin-film silicon PhC solar cells is greater than the numerically predicted 115 efficiency of perovskite-silicon tandem cells and well above the current worldrecord-holding 28.1% tandem cell demonstrated by Oxford PV. 18 These considerations suggest that single-junction thin-film silicon, aided by wave-interference-based light-trapping, may prove to be among the most fruitful avenues for further research and development in next-generation solar cell technology.