Over 65% Sunlight Absorption in a 1 μm Si Slab with Hyperuniform Texture

Thin, flexible, and invisible solar cells will be a ubiquitous technology in the near future. Ultrathin crystalline silicon (c-Si) cells capitalize on the success of bulk silicon cells while being lightweight and mechanically flexible, but suffer from poor absorption and efficiency. Here we present a new family of surface texturing, based on correlated disordered hyperuniform patterns, capable of efficiently coupling the incident spectrum into the silicon slab optical modes. We experimentally demonstrate 66.5% solar light absorption in free-standing 1 μm c-Si layers by hyperuniform nanostructuring for the spectral range of 400 to 1050 nm. The absorption equivalent photocurrent derived from our measurements is 26.3 mA/cm2, which is far above the highest found in literature for Si of similar thickness. Considering state-of-the-art Si PV technologies, we estimate that the enhanced light trapping can result in a cell efficiency above 15%. The light absorption can potentially be increased up to 33.8 mA/cm2 by incorporating a back-reflector and improved antireflection, for which we estimate a photovoltaic efficiency above 21% for 1 μm thick Si cells.


Design parameters and optimisation flow
shows a schematic representation of the optimisation procedure used in this work to achieve the final 3D HUD-based textures. We start with an empty canvas in k-space.
Employing the properties of the Si slab waveguide modes and coupled mode theory (see section Mode Coupling Analysis) we infer a near-optimal diffraction k-range that maximises the absorption efficiency of the incoming solar radiation. In a second step, we populate the k-space range uniformly based on the constraints derived in the previous step. For the HUD patterns, only the inner bound for the wavevector is used.
As the stealthiness parameter χ is increased, a ring distribution naturally forms. Next. we identify the 2D point pattern is obtained whose Fourier transform reproduces the k-space pattern targeted. From the 2D point pattern hence generated, a 2D two-phase structure is obtained in a subsequent decoration step. For the spinodal patterns, the two-phase structure is obtained by the random superposition of cosine waves with random phase with wave vectors imposed by the k-space distribution and thresholding the resulting function at a fixed height value. As a final step, the two-phase pattern is extruded to a height h G and incorporated as part of the full 3D solar cell. Figure S2 is a cross-section representation of the 3D device design that is considered for the FDTD calculations in the last optimisation step. The sames values for h AR , h G and n AR have been fixed to all design. We have used the values as obtained from optimising light absorption in a Si slab with a periodic hexagonal pattern (the periodicity of which was also optimised).
In the final optimisation step of the HUD and spinodal patterned cells, light absorption in the film is calculated at each optimisation step where the average lattice spacing (a) and Si filling fraction (f ) of the patterns is fine tuned. The optimised design parameter values are shown in Table S1.   Figure S3), we deduce the actual thickness of the membrane and resist (listed in Table S2). On the pattern, we expect the ARC layer to deviate from the resist thickness on the unpatterned region.    In our experiments, we have used a polymer resist as the low refractive index material, which has a sub-optimal refractive index (n = 1.52) and thickness (h AR ∼ 50 − 100 nm).
The resulting reflectance (Fig. S5) is on average 10% higher compared to the previous case ( Fig. S4). 6 Absorption spectra in the full device The simulated Si absorption for the optimised HUD-based textures in a full device configuration (i.e. including the Ag back reflector and optimised design parameters listed in Table   S1) are shown in the figure below.

Si and Ag dispersion models
The dispersive dielectric function of c-Si was modeled using using a sum of Lorentzian terms 2 In principle, employing a large number of Lorentzian terms and/or splitting the spectral range 3 for the interpolation can provide an accurate fit for measured of the dielectric constant of silicon. 4 However, the running time for computing the absorption using the simplified Si absorption model for a 16 × 16 × 5µm 3 cell using 300 computing cores and 1.2 Tb of memory, is about 12 hours. Increasing the number of Lorentzian terms and using multiple interpolation spectral ranges dramatically increase the computation time and make the 7 optimization process unpractical. To facilitate the optimisation process, we opt here for a simplified approach and use only two Lorentzian terms in the expansion with the values for the constants involved shown in Table S3.  8 The dielectric function of Ag was modeled using using the parameters from Ref. 2   Table S4 Parameters used to model the Ag dispersion in Eq. 1.

Coupled Mode Analysis
Light-trapping in slabs with thicknesses comparable to the wavelength of the incident light in general is accomplished by coupling to guided optical modes supported by the absorbing layer, which in case of weak absorption have propagation distances typically much longer than the thickness of the slab. The coupling to guided modes is achieved through roughening or corrugating the surface of the absorbing layer, which renders certain portions of the guided modes excitable by incident light, giving rise to guided resonances. 5 For the case of periodic patterns added to the slab's surface, it was shown that the 4n 2 / sin θ-limit calculated for a Lambertian scatterer can be surpassed, when the grated structure exhibits sub-wavelength modal confinement. 6 The analysis is based on temporal coupled-mode theory 7 and applied to the case of nanophotonic light-trapping. The ultimate limit of achievable absorption corresponds to the normalised sum over the absorption spectra of all guided resonances present in the frequency interval of the incident light.
We start with the equation governing for the amplitude of a given single guided reso- with a the resonance amplitude (normalised such that, |a| 2 is the energy per unit area in the slab), ω 0 the resonance frequency, N the number of excitation channels that a given resonance can couple to, γ e the extrinsic loss-rate to each of the N channels, γ i the intrinsic loss rate of the resonance due to material absorption, and S the amplitude of a given excitation channel. Assuming a harmonic expansion for the resonance and the incident wave amplitudes, a(t) = a(ω) exp(iωt), S(t) = S(ω) exp(iωt), the absorption spectrum of the resonance is given by The spectral cross-section of the resonance is found through the integral σ = which, under the assumption of weak variation of N, γ e , and γ i with frequency, is given by In the overcoupling regime γ e ≫ γ i , or in case N ≫ γ i /γ e , the maximal value of Eq. (4) can be approximated as: 6 This maximum spectral cross-section carries the unit of frequency and when normalized by the incident spectral bandwidth ∆ω, it is assumed that the resonance contributes an additional σ/∆ω to the absorptionĀ. The value ofĀ is then found as the sum over the maximum spectral cross-section of all modes, normalised by the incident spectral bandwidth where the index m labels individual resonances and the summation is performed over all modes in the frequency interval ∆ω. When the guided resonance is approximated as a plane wave in the slab, the intrinsic loss-rate can be expressed as Here, κ m and v m and denote the imaginary part of the mode's refractive index and its group velocity, respectively.
We now apply this formalism to the physics of our HUD membranes. Here the modes m are the guided modes supported by the slab (dispersing as in Fig. 2a of the main text), while the channels N m that they are subject to is determined by the structure factor of the surface texture.
The number of channels N associated with a given mode are obtained by considering the structure factor of the surface texture. We approximate the structure factor as a ring in reciprocal space of homogeneous intensity with inner radius k 1 and outer radius k 2 and assume that waves scattered by the surface patterning acquire any wavevector exclusively within this range with equal probability. From the k-space area of the structure factor A = π (k 2 2 − k 2 1 ) relative to the area of a single mode A k = 2πk m ∆k m , the number of channels can then be approximated as  rewriting the same for the narrow band limit ∆ω → c∆k m , we obtain: where superscript ω indicates that α m , k m , and n m are now evaluated at the frequency ω under consideration and not averaged across the resonance's spectral width. Equation 8 clearly highlights the influence of the structure factor on the absorption in the slab. The Furthermore, we conjecture that theÃ ω T is representative of the frequency-dependent absorbance of the film (defined asα(ω) × ℓ, withα(ω) the effective absorption coefficient and ℓ the sample thickness). We consider that this phenomenological approach provides a good starting point in our optimisation process and we emphasise that all designs derived here are further optimised and validated with full-wave 3D FDTD simulations.
In order to calculate an upper limit for the total absorption of light in the slab, we employ the analytical solutions derived for Lambertian light-trapping in textured solar cells, 8 which considers a thin absorber with the texture on the front-, and a perfect metal reflector on the rear surface. The absorption expression is reproduced for convenience below: where the product absorption coefficient times slab thickness αℓ is identified with Eq. (8) as indicated above. The calculated path-length enhancement factor f p = 4 in the weakly absorbing limit results in an over-estimation of the absorption in the small-wavelength range, but is retained nonetheless to facilitate computation of the spectrally averaged absorption.
The latter is found through convolution of Eq. (9) with the AM1.5 photon flux and integration over the range [400, 1050] nm, normalized to the total amount of photons in the same interval: where F (ω) describes the number of photons per square meter and second at the frequency ω as obtained from the AM1.5 spectrum. Figure S9 shows the spectrally averaged absorption as a function of k 1 and k 2 , approximated by Eq. 10; we obtain that the absorption is maximised for k 1 = 9.1 µm −1 and The optimal values for k 1 and k 2 can be further refined by taking into account the 13 Fig. S9 Absorption of a Si slab as a function of the inner (k 1 ) and the outer (k 2 ) radius of the ring-shaped homogeneously distributed structure factor, using the spectrally averaged absorption over the frequency range 400 -1050 nm. Left: Mode data obtained through a finite-difference eigenmode solver for a slab with metal back-reflector in air. Right: Similarly obtained mode data for a slab with back-reflector, anti-reflective coating (h AR = 72 nm, n AR = 1.82) and including Maxwell-Garnett effective medium approximation for the patterned layer.
anti-reflecting coating (h AR = 72 nm, n AR = 1.82) and the corrugated surface by using an effective homogeneous medium with a complex refractive index calculated using Maxwell-Garnett effective medium theory for a slab of thickness 196 nm. Under these assumptions, the intensity is now maximised for k 1 = 9.7 µm −1 and k 2 = 20.4 µm −1 .
For the k 1,2 -values identified above, the effective absorption coefficient absorption,α0(ω)× ℓ is compared with the bulk absorption coefficient times thickness, α 0 (ω) × ℓ, to identify the predicted absorption enhancement across the spectral bandwidth.

Angular dependence of absorption
In this section we contrast the angular dependence of the absorption in the samples analysed. The results presented in Fig. S11b show that the periodic structuring gives rise to a strongly anisotropic absorption, whereas the for a disordered structuring (Fig. S12), be it hyperuniform or spinodal the angular response is mostly isotropic roughly following the unstructured slab (Fig. S11a) case but with an enhanced absorption due to the optimised coupling of the incoming solar radiation to the quasiguided modes of the silicon slab.  corresponds to a dose of 100 µC/cm 2 ). Similar to the experimental results, the HUD network is a very robust design, where the different fabrication parameters leads to a very similar PSD for k<20 µm −1 (i.e. the accessible range in our experiments). By contrast, the PSD for the spinodal design is in particular highly sensitive to fabrication conditions, especially at k<10 µm −1 . In the case of the spinodal design, increasing the dose reduces the filling fraction of Si in the nanotexture and induces the increase of scattering strength below the desired diffraction ring in the wavevectors k < 20µm −1 . We have noted that this is only true in the spinodal design and we believe that it is intrinsic to the nature of the spinodal pattern. Changing the Si filling fraction essentially modifies the cutting threshold ϕ 0 that was used to optimise the pattern.

Device Simulation Parameters in PC1D
To estimate the 1 µm silicon solar cell device performance, we have used the same 1D device model as in the Advanced HE-Tech devices recently reported by Buonassisi's group. 9 Despite the fact that in practice IBC cells require more complex 2D-3D architectures, this 1D model replicates well the performance of existing cells: 26.1% p-type IBC solar cell and 26.3-26.7% n-type IBC solar cells with Si heterojunction HIT architecture. We have also validated our 1D model system by comparing it to 2D simulations for thin IBC cells found in literature.
We find that by using the same parameters in our 1D scheme and modified FSF and BSF doping profiles to thin Erfc functions (5 nm depth factor), we obtain virtually the same PV performance values (See Table S5).  Figure 4 of the main text, are displayed in Table S6. We note however that the effect of Auger recombination is found to be negligible (about 0.01% efficiency decrese) here given the relatively low bulk lifetime.
Therefore, Auger recombination was neglected in the analysis in the following section.
In here, the contact SRV is indicated by the back surface recombination parameter, while the front surface recombination parameter is effectively accounting for recombination at the passivated front surface (modulated by the emitter peak doping and front SRV). The low SRV at the contact is possible through recently reported advances in passivation schemes at the Si-metal contact for high efficiency PV (see for instance Ref. 12). For the front surface, we have used larger SRV values compared to best passivation schemes in flat Si, to account for the complexity of passivating vertical walls in Si. We also extensively discuss the effect of all these parameters in the following section. Moreover, for the three different structures (HUD network, HUD holes and spinodal) the front SRV parameter was multiplied by an SRV factor to account for the additional surface area generated through texturing.
Light trapping by our textures was introduced by tuning the internal reflection parameter in the PC1D model so that J photo matches that from the integration absorption spectra (from 300 to 1050 nm) in the ideal device case (i.e. with minimal recombination losses).
The front external and rear internal (specular) reflectances were fixed to 1.5% and 99.45%, respectively, to account for the average front reflectance and metal (specular) back-relfection in our devices. The internal (diffuse) reflectance was modified equally for first and subsequent bounces. The used internal reflection and a SRV enhancement factors are presented in Table S7, along with the obtained short circuit current (J sc ) open-circuit potential (V oc ) and efficiency (η) for all the different patterning designs. The corresponding I-V and efficiency curves for all four surface texturing are displayed in Figure S14.    Table S6. Figure S15 clearly shows that the efficiency in any of the thin film cells is hardly affected by increasing the bulk recombination lifetime from 100 µs to 2000 µs. This observation is 21 explained by the fact that a 1 µm silicon membrane is too thin for bulk recombination to play a major role in recombination losses. Bulk lifetime thus has little to no influence on thin film Si and bad quality silicon can therefore be used for thin films without a major loss in efficiency.   Table S6. For the sake of comparison, we have also considered the same light trapping and enhanced surface area given by the different nanopattern designs. Because of the small effect of SRV to the total efficiency in bulk Si, the V OC is the same (within two decimal spaces) for all the designs.    Integrating sphere set-up