Optimization-based design of surface textures for thin-film Si solar cells — Are conventional Lambertian models relevant?

The efficiency of thin film silicon solar cells critically depends on the optical absorption. In this work, we numerically investigate the light trapping effect in the weak absorption regime for 1.5 μm thick crystalline silicon at normal incidence. Using electromagnetic simulations, we study the impact of different textures on the light absorption enhancement in two-dimensional (2D) device configuration. We discover that the light trapping effect of commercially used textures from Asahi glass is close to the 2D Lambertian result (F = πn), while optimized periodic gratings show better performances, and both rectangular and triangular gratings are capable to exceed the Lambertian result. To further explore the optimal light trapping structures, we develop a simulation method combining finite-difference time-domain (FDTD) calculations and the optimization algorithm (Nlopt-COBYLA). Over a 900–1100 nm wavelength range, our optimized surface texture in two dimensions (2D) enhances absorption by a factor of 2.7πn, considerably larger than the original πn Lambertian result and exceeding by almost 50% a recent generalization of Lambertian model for periodic structures in finite spectral range. However, our structure still obeys the conventional πn Lambertian limit when averaged over all the angles.


Introduction
In this paper, we use numerical optimization of surface textures to improve the efficiency of thin-film silicon photovoltaics-a promising technology for solar energy due to their low-cost large-scale manufacturability [1,2], but for which a major challenge is to absorb light at infrared wavelengths where the absorption length is much larger than the active layer thickness. Previously we and others proposed and experimentally demonstrated the effectiveness of using deterministic periodic structures as promising solutions for efficiency enhancement [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18]. Front and/or back surface textures (whether periodic or aperiodic)  can improve efficiency by scattering normally incident light into the plane of the film, but are limited by the fact that the same texture also scatters in-plane light back out of the film. In order to understand these limits, various authors have considered textured-surface absorption under models with restricted assumptions for which an explicit limit can be derived. The most well-known models are the Lambertian models [26], which consider surface textures under the key assumptions of isotropic scattering, in which case the efficiency enhancement is at most πn in 2D or 4n 2 in 3D for a film (with index n) of infinite thickness and spectral range. More recently Yu et al. [24] showed that for a thin film over a wavelength range from λ 1 to λ 2 the best Lambertian enhancement is increased to πn(1 + λ 1 /λ 2 ) for 2D periodic structures at normal incidence. It is known that higher efficiency can be achieved in angular-sensitive devices [29], however, and we show how this enhancement/sensitivity tradeoff can be optimized while remaining in the context of periodic thin-film textures. In particular, we consider normally incident light on a Si thin film, backed by a textured SiO 2 layer and a reflecting mirror as shown in Fig. 1, in two dimensions for simplicity. A large amplitude of surface texture will not be an isotropic scatterer, and we show that even simple periodic textures like the sawtooth grating can exceed Yu's generalized Lambertian results. We also show that symmetric gratings, like the triangular grating, lose approximately a factor of 2 in performance as predicted by Ref. [24], and we show that considerably worse performance is achieved by random texture of commercial Asahi glass [27]. We apply multi-parameter optimization to a general texture, described by an arbitrary Fourier series, to obtain the best performance to date: enhancement 50% larger than Yu's limit (and even larger efficiency may be possible as the particular set of optimized parameters only represents a local optimum rather than a global one). However, our structure still obeys the fundamental Lambertian limit derived for isotropic input light [29], and in that sense optimizes the tradeoff between enhancement and angular sensitivity for periodic textures.
Lambertian models were pioneered by Yablonovitch [26], who considered absorption in a thick slab (d >> λ) by surface texturing in an ergodic ray-optics regime over infinite spectral range, and showed that a 4n 2 limits on absorption results in 3D, and a similar result was shown by Stuart and Hall [28] under an isotropic scattering assumption in the wave optics regime. In 2D, the corresponding limit was shown to be πn [24]. Specifically, for a film of thickness d and absorption coefficient α, the dimensionless absorption enhancement factor F is defined to be:  (1) where A(λ) is fraction of light absorbed by the structure at each wavelength, averaging over a range from λ 1 to λ 2 . The 4n 2 and πn results were recently extended to normal incidence on periodic textures with finite thicknesses and finite spectral range by Yu et al. [24], who modeled the film by a set of guided-wave resonances; they assumed isotropic scattering by assuming that each resonance radiates equally into all the channels. (They also implicitly assumed weak scattering, in that the leaky-mode description of resonances is only valid if the resonances decay slowly. On the other hand, they assumed that the radiative scattering occurs much more quickly than absorption). They showed that the πn limit in a periodic 2D structure could be increased by a factor of 2 at a single wavelength under these assumptions, and when their result is averaged over a finite spectral range one obtains a πn(1 + λ 1 /λ 2 ) enhancement for an -optimal‖ grating period Λ = λ 1 . This result in their model is achieved for asymmetric textures like the sawtooth grating in Fig. 1, whereas they argue that symmetric textures (like the triangular grating in Fig. 1) lose a factor of 2 in efficiency because normal-incident light can only couple to half of the guided-wave resonances (the symmetric resonances) [4,24].  However, it is known that these 4n 2 and πn factors are not fundamental limits if one considers incident light over only a narrow range of angles and wavelengths. More fundamental Lambertian limits were proved for the case of isotropic incident light-in this case, without any restriction to isotropic scatterers, it was proved that the angle-averaged enhancement is at most πn in 2D and 4n 2 in 3D, assuming that the absorbing layer is thick enough to be described by the local density of states of the uniform material [29]. If the incident angles are restricted to lie within [-θ, θ], then it was argued that the enhancement bound is instead 4n 2 /sin 2 θ (for 3D) or πn/sinθ (for 2D) [34]. If one is willing to restrict consideration to a limited range of angles, therefore, the question is then how one can achieve this enhancement in practice. Over a very narrow spectral range, one can use planar (nontextured) surfaces to design a resonant mode achieving nearly 100% absorption (arbitrarily large enhancement as the absorption coefficient vanishes) by Q-matching (impedance matching) [30][31][32][33], although this requires a spectral range of at most the inverse of the absorption lifetime. Over a much larger spectral range, one could conceivably use a large lens to concentrate the incident light onto a small spot, an idea used in various solar concentrator designs [34]. On the other hand, the situation is less clear if one has a large spectral range but restricts oneself to wavelength-scale periodic textures, in order to have translation-invariant performance (unlike a lens), e.g. to enable large areas or to have a very thin device. One can certainly design periodic or other textures to respond resonantly to normal-incident light, but as the incident angle is changed one would normally expect these resonances to shift continuously in frequency, so that many resonances will still lie within the bandwidth over some range of angles. To optimally exploit a narrow range of incident angles, while retaining a relatively large spectral range and a periodic texture, one needs to somehow achieve resonances that not only shift rapidly with angle but also rapidly alter their lifetimes or amplitudes. One therefore clearly needs a texture that scatters very anisotropically so as to be angle-sensitive, in which case the isotropic Lambertian models for normal-incident light no longer apply. In order to fully explore these tradeoffs in the absence of analytical Lambertian models, one ultimately requires large scale optimization to explore the large space of texture parameters.

Computational method
The 2D device structure used in our numerical model is illustrated in Fig. 1. From top to bottom, it consists of air (semi-infinite), 1.5 μm crystalline silicon (Si), 0.5 μm silicon dioxide (SiO 2 ) and a perfect reflector on the backside. For the sake of simplicity, electrical components like metal grids or transparent conductive oxides (TCOs) are not included in the optical model. Meanwhile, the silicon is considered to be intrinsic and the effects of p and n regions are neglected. The front surface of silicon and the interface between SiO 2 and reflector are kept flat, and only the interface between Si and SiO 2 are textured. It should be noted that the volumes of Si and SiO 2 are kept constant when different type of textures are evaluated. That means, if the texture is characterized by a function H(x), we enforce that the average H avg is held constant at 0.5 μm.
To simulate this structure, we employ a finite-difference time-domain (FDTD) method [35] implemented via an open source code Meep [36]. The simulation cell size is Λ in the x direction, with a periodic boundary condition for normal incident light and a grid resolution of 10 nm. The material refractive indices are 3.6 for Si and 1.4 for SiO 2 . The Si layer is assumed to be weakly absorptive, with a constant absorption coefficient (α = 12.56 cm 1 ) between 900 nm and 1100 nm (implemented as a constant conductivity). In this wavelength range, the unit cell is illuminated under normal incidence by TE polarized light (with the electric field perpendicular to the plane) in the spectral range from 900 nm to 1100 nm. The absorption spectrum A(λ) is calculated by Fourier-transforming the response to a short pulse to obtain A = 1reflection, with a wavelength resolution of 1 nm. The performance of the texture is characterized by the dimensionless enhancement factor F of Eq. (1), which is the averaged absorption divided by the single-pass absorption αd. Different light-trapping structures are simulated and compared in terms of their performance F. For this wavelength range (from 900 nm to 1100 nm), Yu's model [24] predicts an optimal F of 1.8πn for Λ = 900 nm with asymmetric textures, and an optimal F = πn for Λ   with symmetric textures. This theory is verified by simulating a shallow grating in our model.

Simulation results
2D model for comparison, 1D textures were extracted from the measured AFM image and imported into the model mentioned in Fig. 1. To obtain a more accurate result, 20 different 1D textures (with a length of 4000 nm) were arbitrarily selected and simulated, and we computed enhancement factors F from 0.90πn to 1.28πn (average (1.04 ± 0.10)πn), which are very close to the prediction of a conventional 2D Lambertian surface (F = πn) [24].

Light trapping by symmetric triangular gratings
Diffractive gratings have been proposed as promising candidates to improve solar cell efficiency [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18]. Here we implement a symmetric triangular grating into our model (Fig.  2a), and investigate its influences on light trapping. Absorption-enhancement factors F are calculated based on the simulated absorption spectra (from 900 nm to 1100 nm), and plotted as a function of the grating period Λ and thickness t. As shown in Fig. 2b, the optimal regions are around Λ = 900 nm, although there also are other parameters shown to have high absorption. Figure 2c shows the absorption spectrum of the optimal structure indicated in Fig.  2b (Λ = 920 nm and t = 520 nm). The calculated maximum F is 1.26πn, which exceeds the value πn predicted by the Yu's model for symmetric gratings [24].

Light trapping by asymmetric sawtooth gratings
We now investigate the performance of an asymmetric sawtooth grating (Fig. 3a), following the inspiration that asymmetry should enhance performance for normal incidence [4,24]. Similar to Section 3.2, the enhancement factor F is plotted as a function of Λ and t in Fig. 3b. Optimal regions are still around Λ = 900 nm; however, F is significantly higher than that obtained for symmetric grating, with a maximum value of 2.04πn. Again, this result exceeds the value 1.8πn predicted by Yu's model for asymmetric gratings [24]. The optimal structure has Λ = 920 nm and t = 240 nm. As illustrated in Fig. 3c, the absorption spectrum for the optimized sawtooth grating has many more resonance peaks than the optimized symmetric grating (Fig. 2c), which causes a larger enhancement. This is because in structures with mirror symmetry (Fig. 2a), normally incident plane wave can only couple to even modes while resonances with odd modes remain unexcited [4,24].

Light trapping by grating with optimized Fourier series
To further explore the light trapping limit of a periodic structure, an arbitrarily shaped texture should be studied. In principle, any arbitrarily shaped periodic texture, described by the function H(x) (in nanometers), can be expanded in terms of its Fourier series: where 500 nm is the averaged thickness of SiO 2 layer (the volume of SiO 2 is fixed), and Λ is the period of the simulation cell. To ensure that the texture does not cross the SiO 2 / mirror interface, we use a constraint |H(x) -500| < 500. Because the absorption can be numerically calculated for any given device texture H(x), the enhancement factor F is directly related to the Fourier coefficients (A 1 , B 1 , A 2 , B 2 , …) and period Λ:

., ) F F A B A B 
(3) Therefore, we can optimize the device performance F by tuning the parameters (A 1 , B 1 , A 2 , B 2 … and Λ). As a practical matter, the number of parameters is limited by the simulation capability. We choose the first 5 orders as degrees of freedom, i.e. from (A 1 , B 1 ) to (A 5 , B 5 ), and set higher-order Fourier coefficients to zero. One of those parameters is redundant for periodic boundary condition, because it corresponds to a phase shift. Therefore, we can set B 5 = 0 without loss of generality. Based on previous results in Section 3.2 and 3.3 as well as the prediction of Yu's model [24], we expect that the optimal structure has a period of around 900 nm, so we choose the initial Λ to be 900 nm, but Λ is permitted to be varied in the optimization. The FDTD method is combined with a nonlinear optimization toolbox NLopt [37]. The algorithm we use is based on constrained optimization by linear approximation (COBYLA) [38]. Generally, this problem may have many local optima, and it is infeasible to guarantee that the global optimum has been obtained, so these local optima only represent a lower bound on the attainable performance. We also explored the use of a genetic globaloptimization algorithm [39], but its convergence rate was so slow that over feasible run times it obtained inferior results than repeated local optimization. We run the optimization several times with different initial parameter values to explore different local optima. In 3 runs, we achieve optimized enhancement factors (F) of 2.07πn, 2.10πn and 2.28πn, respectively. These local optima are noticeably larger than the prediction of the Yu's model [24]. In addition, the optimizations keep Λ close to its initial value of 900 nm.
In our model, textured Si/SiO 2 interface is the critical component introducing strong anisotropic scattering, while the magnitude of the surface roughness is determined by the thickness of the silicon oxide layer. Therefore, to further explore the light trapping limit, we introduce the oxide thickness t ox as an additional parameter for optimization. With this additional parameter, we achieve optimized enhancements F of 2.28πn, 2.38πn and 2.70πn, which are even higher than the obtained results when t ox was fixed at 500nm in the previous simulations. The best found parameters are listed in Table 1, with a calculated F = 2.70πn, which is 50% larger than the prediction of the Yu's model (F = 1.8πn). Of course, it is possible that other local optima exist with even better performance.
The above method explores light trapping performances for asymmetric structures, since it includes both sine and cosine coefficients. By using only cosine coefficients, we can also optimize symmetric structures. Here we intentionally set all the sine coefficients (A 1 to A 5 ) to zeros. We obtain optimized F of 1.56πn, 1.45πn and 1.36πn, respectively. The performances of optimized symmetric structures are much lower than those achieved for asymmetric structures, but still larger than the prediction of the Yu's model [24] and are also larger than the performance of the optimized triangular grating.

Comparison and discussion
Our simulation results are compared with the generalized Lambertian model by Yu et al. [24]. As mentioned in the introduction, the enhancement factor F for 2D period-Λ structures in this Lambertian model is:  (5) In Ref. 24, F is plotted for a single wavelength λ, whereas here we average over a given range from λ 1 = 900 nm to λ 2 = 1100 nm. As shown in Fig. 4 (in green), this maximum F occurs when Λ = 900 nm, in which F = 1.8πn for asymmetric structures. For symmetric structures, F = 0.90πn for the first maximum, but approaches πn as Λ goes to infinity. As we mention in Section 3.1, textures from commercial Asahi glass (the dotted line in black) show performance close to πn.
Performances of our simulated triangular and sawtooth gratings are also shown in Fig. 4 (in blue). Here we select and plot the largest F at each period Λ in Fig. 2b and Fig. 3b. As illustrated, both triangular grating and sawtooth grating follow a trend similar to the analytical models, peaking at around Λ = 900 nm. However, these simulated results F deviate from the Lambertian models, showing higher values for most periods.
In addition, we illustrate the results for gratings with optimized Fourier series in local optima for several runs with different initial values at Λ 900 nm. Nevertheless, all of them exceed the optimal sawtooth gratings as well as the Lambertian models. The best Fourier structures are also illustrated in the insets, with F = 2.70πn for asymmetric structures and F = 1.56πn for symmetric structures. These results clearly demonstrate that violating the isotropic coupling assumption in Yu's model [24] can lead to higher performances.

Angular response of the optimized structure
As explained in the introduction, this super-Lambertian enhancement for normal incident light must necessarily come at the expense of enhancement at other angles. To demonstrate this, in Fig. 5 we plot the enhancement factor versus incident angle, as computed by a rigorous coupled wave analysis (RCWA) method [6]. As expected, the enhancement factor F decreases significantly for non-normal incidence [24]. F is larger than πn for incident angles within about [-25°, 25°]. The average enhancement factor over all angles π 2 2 1 ( ) cos 2 is calculated to be 0.9πn, which still obeys the classical Lambertian limit for isotropic light [26,29]. It should be noted that low-cost thin-film Si solar cells are usually implemented within a non-concentrated configuration like a roof-top system, so strictly restricting the angle range is not practical. Nevertheless, our design provides a methodology for achieving high efficiency solar cells with restricted angles. In addition, our design is still instructive for practical applications since the incoming power of sunlight is not isotropic and usually peaks in the middle of the day. Therefore, it is still meaningful to design a cell with a better performance for normal incidence.

Conclusion
Lambertian models provide a simple and instructive intellectual framework for describing the effects of surface texturing on thin-film absorption. Even outside their range of validity (isotropic scattering), they can still be surprisingly descriptive: in our case, we find that the Lambertian prediction gives a rough guideline as to the optimal texture period, and is within 50% of the actual performance. In order to design an optimal texture structure at normal incidence, including strong anisotropic scattering, one must eventually depart from the confinements of analytical models, however beautiful, and resort to brute-force computation and parameter optimization. The results in this paper provide a glimpse of what is possible from such a computational approach, in which significant improvements are obtained even by a small number of local optimizations (despite the nonlinear and nonconvex nature of this optimization problem). Such local optima represent only lower bounds on the attainable performance, and by expending additional effort one could certainly envision pushing those bounds upwards, although locating the true global optimum with confidence seems daunting. It may also be possible to analytically prove more general upper bounds on performance for gratings with a given period and specific incident angle. As a practical matter, a more important goal is to adapt these techniques to 3D. In order to improve the efficiency of the calculation and make 3D optimization feasible, a number of techniques could be employed. First, one could use more sophisticated computational techniques than FDTD, such as boundary-element methods that only require the interfaces to be discretized [40]. Second, one could use adjoint methods to compute the gradient of F with respect to the optimization parameters [41], and thereby employ much more efficient gradient-based optimization methods (such methods have been used in topology optimization of photonic structures with hundreds or thousands of degrees of freedom [42][43][44][45][46]). As a heuristic method, it might be interesting to investigate using the optimized Fourier coefficients from the 2D simulations in this paper to form a two-dimensional texture with similar frequency components in 3D.