Topology optimization for light-trapping structure in solar cells
Shuangcheng Yu, Chen Wang, Cheng Sun & Wei Chen
Structural and Multidisciplinary Optimization ISSN 1615-147X Struct Multidisc Optim DOI 10.1007/s00158-014-1077-z
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Author's personal copy Struct Multidisc Optim DOI 10.1007/s00158-014-1077-z
RESEARCH PAPER
Topology optimization for highly-efficient light-trapping structure in solar cells Shuangcheng Yu · Chen Wang · Cheng Sun · Wei Chen
Received: 29 May 2013 / Revised: 5 February 2014 / Accepted: 22 February 2014 © Springer-Verlag Berlin Heidelberg 2014
Abstract The limitation associated with the low optical absorption remains to be the main technical barrier that constrains the efficiency of thin–film solar cells in energy conversion. Effective design of light-trapping structure is critical to increase light absorption, which is a highly complex phenomenon governed by several competing physical processes, imposing a number of challenges to topology optimization. This paper presents a general, yet systematic approach exploiting topology optimization for designing highly efficient light-trapping structures. We first demonstrate the proposed approach using genetic algorithm (GA) based non-gradient topology optimization (NGTO), which is robust for achieving highly-efficient designs of slot-waveguide based cells with both low-permittivity and high-permittivity scattering material at single wavelength or over a broad spectrum. The optimized light-trapping structure achieves a broadband absorption efficiency of 48.1 % and more than 3-fold increase over the Yablonovitch limit. The fabrication feasibility of the optimized design is also demonstrated. Next, the gradient topology optimization (GTO) approach for designing light-trapping structure is explored based on the Solid Isotropic Material with Penalization (SIMP) method. Similar designs are obtained through both GA based NGTO and SIMP based GTO, which verifies the validity of both approaches. Insights into the application of both approaches for solving the nanophotonic design problem with optimization nonlinearity are provided.
S. Yu · C. Wang · C. Sun · W. Chen () Department of Mechanical Engineering, Northwestern University, Evanston, IL, USA e-mail:
[email protected]
Keywords Topology optimization · Light-trapping structure · Nanophotonic design · Genetic algorithm · SIMP · Solar cell
1 Introduction Organic solar cells promise an economically viable sustainable energy source due to their potential in low-cost production (Shah et al. 1999). As the diffusion length of organic active material is typically 10–20 nm, thin active layer is favorable for organic solar cell (Shaw et al. 2008). Yet the photon absorption length of the active material is an order of magnitude larger than the diffusion length (Park et al. 2009). A significant reduction of thickness of the active layer will result into a low optical absorption, hence a poor power conversion efficiency (Shaw et al. 2008). Light trapping technology was therefore developed (Yablonovitch 1982; Polman and Atwater 2012) to extend the path-length for light interacting with the active layer, which can boost the absorption to offset the reduction of the active layer. Recent advances in nanophotonics have opened up a new gateway for light trapping (Yu et al. 2010; Callahan et al. 2012) with enhancement of absorption exceeding the classical Yablonovitch Limit (Yablonovitch 1982). Light trapping in thin-film solar cells is governed by several competing physical processes, including light refraction, deflection, and absorption. Pursuing the optimal structure requires a comprehensive consideration of all these competing processes. A wide range of structures have been investigated by using forward design methods, such as the triangular or pyramid grating (Dewan et al. 2009; Feng et al. 2007), nanowires (Garnett and Yang 2010), nanoholes (Raman et al. 2011), nanocone (Wang
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et al. 2012), nanoparticles (Chen et al. 2012), and other nanostructures (Bermel et al. 2007; Mallick et al. 2011; Atwater and Polman 2010; Schuller et al. 2010; Wang et al. 2010; Aydin et al. 2011). However, these designs are conducted in an ad-hoc fashion that relies on physical intuition to predefine the topology of the light-trapping structure, and thus, not capable of handling the topological variation. This fundamentally constrained design space cannot drive the optimization to reach the optimal design with superior performance. Therefore, there is a need for a design methodology that is capable of searching the optimal topology in delivering highly efficient light-trapping structure. Topology optimization, originally developed for mechanical structure design problems (Bendsøe and Sigmund 2003; Bendsoe and Kikuchi 1988; Rozvany 1992) and recently extended to the design of photonic devices (Jensen and Sigmund 2011), provides a promising solution of searching for the optimal light-trapping structure in thin-film solar cells. In the past decade, various high-performance nanophotonic device designs have been obtained using topology optimization, such as the photonic crystal structure with maximum band-gap (Kao et al. 2005; Preble et al. 2005; Shen et al. 2003), low-loss photonic waveguide (Jensen and Sigmund 2004), photonic structure for light confinement (Gondarenko et al. 2006), and the invisible cloak (Andkjaer and Sigmund 2011). Nevertheless limited effort has been made to achieve efficient nanophotonic light trapping structures for thin film cells by using topology optimization. Miller et al. applied shape optimization approach to seek efficient nanophotonic light trapping structure by decomposing the active/dielectric material interface profile into Fourier basis functions (Miller et al. 2012). Similarly, Soh et al. optimized the active/dielectric material interface using topology optimization approach (Soh and Yoo 2012), whereas the design is only for totalinternal-reflection (TIR) scheme. Fabrication constraint is not considered in both works, which lead to infeasible designs for fabrication. Later, Soh et al. obtained a traditional multilayered structure as the optimized design for thin-film silicon solar cells [32]. Instead of directly targeting at the cell absorbing performance, the optimization in their work is formulated to maximize the energy transmitted into the active layer. Limited improvement is found in their designs. As topology optimization being applied to different applications, various topology optimization methods have been developed since the seminal work of Bendsøe and Kikuchi (Bendsoe and Kikuchi 1988). Based on whether gradient information is used in searching a solution, topology optimization methods can be generally categorized into two classes, i.e. nongradient-based topology optimization (NGTO) methods (Wang and Tai 2005; Shim
and Manoochehri 1997; Luh and Lin 2009) and gradientbased topology optimization (GTO) methods (Sigmund 1997; Luh and Lin 2009; Sigmund 1997; Wang et al. 2003; Bendsøe 1989; Zhou and Rozvany 1991; Bensoe and Sigmund 1999). Sigmund made a comprehensive comparison between NGTO and GTO methods from a theoretical perspective (Sigmund 2011). It was noted that GTO methods outperform NGTO methods in solving most optimization problems, but GTO may be less effective for some special applications, such as certain nanophotonic design problems with numerous local optimums (Wang et al. 2013) or disjoint design space (Sigmund and Hougaard 2008). However, there is limited research on solving these special problems using topology optimization due to the complexities in physical modeling and gradient evaluations. Meanwhile, a comparison between the performances of NGTO and GTO approaches in dealing with these non-conventional problems is worthy of investigation. Considering the aforementioned challenges in efficient light-trapping structure design, we propose a general, yet systematic approach based on topology optimization for achieving highly efficient nanophotonic light-trapping structures with simultaneous consideration of all competing governing factors over light absorption. This approach can be readily applied to solar cell devices with different light-trapping schemes. The optimized design is feasible for fabrication with existing nanofabrication techniques (Rai-Choudhury 1997). While some preliminary results using our method was published (Wang et al. 2013), an in-depth description and examination of the topology optimization approach is provided in this paper. Both GA based NGTO (Wang et al. 2013) and SIMP (Jensen and Sigmund 2011) based GTO are explored to draw insights into the usefulness of these two approaches in solving the nanophotonic design problems with nonlinearity. This paper is organized as follows: A brief introduction to light trapping process, the formulation of the light-trapping structure optimization, and the Rigorous Coupled Wave Analysis (RCWA) as the forward analysis method are presented in Section 2. After that, the GA based NGTO approach for efficient light-trapping structure design is presented in Section 3 with two demonstrative examples. In Section 4, the SIMP based GTO approach is explored to designing efficient structures using two different scattering materials. Conclusion and future work are discussed in the last section.
2 Technical background 2.1 General introduction to light-trapping process The theory of light trapping was initially developed on the basis of total internal reflection for conventional semi-
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conductor solar cells, where the active layer is typically many wavelengths thick (Yablonovitch 1982). By texturing the interface between the cell and air, the light propagation inside the active layer is randomized, leading to a much longer propagation distance and hence an absorption enhancement. For such light-trapping schemes, the upper limit of the absorption enhancement is analyzed by statistical ray optics to be 4n2 /sin2 θ (Yablonovitch 1982), where θ is the angle of the emission cone in the medium surrounding the cell and n denotes the refractive index of the active material. This limit is also known as the Yablonovitch limit. For nanoscale light-trapping schemes with the thicknesses of active layer comparable to or even smaller than the wavelength, some basic assumptions of the conventional theory are no longer applicable. It has been shown that nanophotonic light-trapping schemes possess the potential to achieve enormous optical absorption enhancement surpassing the Yablonovitch limit (Yu et al. 2010; Callahan et al. 2012) Various nanophotonic light-trapping schemes with sub-wavelength structures have been proposed, including dielectric structures built at the top of the cell (Yu et al. 2010; Raman et al. 2011), photonic crystal structures in the active material itself (Mallick et al. 2010), or metallic nanostructures constructed in the active layer or on the interface between different layer (Atwater and Polman 2010; Aydin et al. 2011). In these cases, high absorption can be achieved by coupling the incident light into a few distinct modes existing in thin film cells. Random scattering pattern is no longer optimal as it couples incident light into continuous modes. Therefore, we aim at a systematic design approach to pursuing the optimal periodic structure designs for nanophotonic light trapping. 2.2 Formulation of the light-trapping structure optimization problem In this work, the goal of optimization is to maximize the optical absorption in the thin-film cells. For example, Fig. 1 a and c present a general structure of a thin-film solar cell model, where the structure on the first layer, i.e. the scattering layer, is the main light-trapping structure to be optimized for maximal absorption in the active layer. The design problem is then recast as the optimization of the material distribution, hence the permittivity distribution ε, within the design space, i.e. the unit cell of a periodic structure shown in Fig. 1b. The design objective is to optimize the absorbing performance of the thin-film cell, represented by the absorption coefficient A. Denoting the absorbed portion of the total incident energy, A is the most direct and complete metric judging the absorbing performance of thin-film cells. The
formulation of this nanophotonic light-trapping structure optimization problem is shown as (1): I − R (ε) − T (ε) − D (ε) ; (1) ε I Where I is the incident energy, R and T are the energy of the zeroth -order reflection and transmission respectively, D denotes the deflected energy that considers higher order reflection and transmission, and ε represents the permittivity distribution. Different from other conventional problems, there is no volume constraint in this optimization. Nevertheless, the minimum length-scale of the design needs to be controlled for fabrication ease, which can be attained by applying filtering (Sigmund 2007). max A (ε) =
2.3 Rigorous coupled wave analysis To evaluate the absorption coefficient A in a thin-film solar cell, the Rigorous Coupled Wave Analysis (RCWA) method (Li 1997; Moharam et al. 1995a) is employed. RCWA solves the Maxwell’s equations in Fourier domain, which is more computational efficient than time domain solver for dealing with periodic structures (as shown in Fig. 1a) with plane wave incidence. For a multi-layered photonic system, Fourier expansions in each layer of both the field and the permittivity lead to an algebraic eigenvalue system in each layer. The number of Fourier components considered in the optimization process is 13× 13 = 169, where 13 is the total diffraction order chosen in each direction in the analysis. The convergence test was performed on the selection of the diffraction order to ensure the numerical accuracy. As the scattering layer is discretized by a N x × N y mesh, the Fourier components of the permittivity can be calculated using (2): ε kx , ky =
Ny Nx 1 ε xi , yj e−i·(kx xi +ky yj ) (2) Nx × Ny i=1 j =1
The electromagnetic fields in each layer are determined by solving the corresponding eigen-problem. The scattering matrix (S matrix) algorithm (Li 2003) is then applied to integrate multiple layers across the whole configuration. After that, the Poynting theorem is implemented to calculate reflection, transmission, and deflection. The absorption coefficient is evaluated following the law of energy conservation shown in (3): A + R + T + D= I ; D= Rn + Tn n =0
(3)
n =0
where I is the incidence, A is the absorption, R is the zeroth-order reflection, T is the zeroth-order transmission, and D is deflection, which includes higher order reflection and transmission. Literature such as (Li 1997;
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Fig. 1 Nanophotonic light-trapping structure illustration
Moharam et al. 1995b) is recommended for more detailed introduction on RCWA.
3 GA based NGTO for light-trapping structure optimization 3.1 GA based NGTO method Genetic Algorithm (GA) is a stochastic search algorithm mimicking the natural evolution process (Goldberg 1989; Holland 1975). It has been extended to topology optimization as a search algorithm in different applications (Preble et al. 2005; Shen et al. 2003; Gondarenko et al. 2006; Wang and Tai 2005; Wang et al. 2006). Using the bitarray structure representation, GA provides a natural way for the 0/1 integer programming that is favored by topology optimization problems with discretized elements (0-no material; 1-with material). Following the idea of survivalof-the-fittest, GA iteratively applies probabilistic operators including selection, crossover, and mutation to search for the optimum; the algorithm is robust to highly nonlinear design problem (Goldberg 1989). In this work, we propose a GA based NGTO approach to optimize the nanophotonic light-trapping structure for thin-film solar cells; procedure is shown in Fig. 2. Since the invention of GA method in 1975 (Holland 1975), a wild array of variants of the original GA methods have been developed as a search algorithm for global optimization. In our work the standard GA (Goldberg 1989) is used, which has been successfully applied to designing photonic crystal and light-confinement structures (Shen et al. 2003; Gondarenko et al. 2006). The
absorption coefficient value of each structural configuration is evaluated using the RCWA forward analysis and treated as the fitness value. We use the tournament selection scheme that has been proved to be more favorable than others (Goldberg and Deb K 1991), and the elitism scheme (Reed et al. 2000) is adopted to preserve the best design within the current generation. The uniform crossover scheme, shown to be effective for 2D topology optimization (Shen et al. 2003), is then used to perform a structured, yet randomized
Fig. 2 GA based NGTO
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exchange of topological traits. In the mutation operation that follows, the value of design variable of each element may be switched to their opposite phase at a very small probability for potential local refinements that are hard to be achieved through crossover. As a gradient-free algorithm, GA suffers the curse of limited design dimensionality (Sigmund 2011), considering the infinite design dimensionality of topology optimization. However, in designing a nanophotonic light-trapping structure for thin-film solar cells, the design dimensionality is significantly reduced compared with conventional topology optimization problems. First, since the solar cell devices operate under the sunlight at all polarization angles, symmetrical scattering pattern is preferable for the optimal light-trapping effect (Yu et al. 2010). Second, with the consideration of strong scattering effect, the periodicity is fixed comparable to the wavelength of visible light. In this work, the D4 symmetry constraint is imposed on the 600 nm × 600 nm unit cell discretized with a 64 × 64 mesh. The design space is reduced to the triangular area denoted in Fig. 1b to facilitate the GA based NGTO in seeking the effective light-trapping structure feasible for fabrication. As a stochastic algorithm, GA renders the issues of numerical instability (Sigmund and Peterson 1998) and disconnected features in solving conventional mechanical structure design problems (Wang and Tai 2005). A discontinuous Heaviside density filter is applied in our work to alleviate the numerical instabilities in GA based NGTO. The gradient-free characteristic of GA and the absence of volume constraint enable the use of the true Heaviside step function in the filter. The design variables x valued either 0 or 1 are first mapped onto the element space by computing the distance-weighted average following the conventional density filtering process (Sigmund 2007). After that, a true Heaviside step function with the threshold value of 0.5 is applied to achieve a pure 0/1 design. It should be noted that the element pseudo density ρ through the discontinuous Heaviside density filter carries the physical meaning (Sigmund 2007). The structure evaluated by the RCWA is the permittivity distribution ε determined by ρ, whereas the GA operators, i.e. selection, crossover and mutation, are performed on the designs represented by x. In this way, the filtering process not only suppress the numerical instabilities associated with the GA based NGTO, but applies a control over the feature length scale for fabrication ease as well. While the aforementioned element disconnection issue caused by GA based NGTO is often undesirable (Wang and Tai 2005), the disconnected features are common and also acceptable in the nanophotonic devices optimization (Kao et al. 2005; Sigmund and Hougaard 2008). With the GA based NGTO framework determined, a proper set of control parameters in GA is critical for a competent optimization. The 3-step methodology (Reed et al.
2000) is followed in this work to determine the values of GA parameters. The population size N p , i.e. the number of designs within one GA generation, is selected to be 2 × N (N is the number of design variables). The total number of evolving generation N g is set as 1.4 × N p and the elitism scheme (Reed et al. 2000) is adopted to preserve the best design within the present generation to the next generation. The parameter s for the tournament selection scheme, denoting the number of candidate designs in tournament selection, is chosen to be 2. The values of crossover probability Pc and mutation probability Pm are thus chosen to be (s-1) /s and 1/N p , respectively. 3.2 Nanophotonic light-trapping structure optimization results using GA based NGTO The proposed GA based NGTO approach for nanophotonic light-trapping design is applied to the slot-waveguide based thin-film solar cell shown in Fig. 1c. Two different materials for the scattering layer are tested as two examples with differing physical characteristics. In these examples, a normal incidence at π/4 polarization angle is applied and a D4 symmetry constraint is imposed on the unit cell. 3.2.1 Light-trapping structure design using GA based NGTO for slot waveguide based solar cell with low-permittivity scattering material Recently, a promising nanophotonic light-trapping scheme (Yu et al. 2010) is investigated utilizing the so called slot-waveguide effect (Almeida et al. 2004). A slot waveguide is an optical waveguide that can confine light in a sub-wavelength region of lower permittivity bounded by two higher-permittivity claddings. However, there exists momentum mismatch between incoming light and intrinsic modes in slot waveguide. Without a properly designed front scattering layer, the mismatch will lead to inefficient coupling of the incident light and lower absorption. A maximum absorption enhancement exceeding Yablonovitch limit at a single wavelength by incorporating sub-wavelength dielectric grating scattering pattern on the cell front is reported (Yu et al. 2010). However, the optimal pattern designing has not been systematically investigated. Here we use this thin-film solar cell to demonstrate the GA based NGTO approach in achieving highly-efficient light-trapping structure. As shown in Fig. 1c, the test structure of slotwaveguide based cell consists of an ultrathin P3HT:PCBM active layer of 10 nm thickness, which is sandwiched between two 100 nm thick cladding layers made of highpermittivity gallium phosphide (GaP, ε = 12.75), and a nano-structured light scattering layer on top. In this example, the light-trapping structure to be designed, i.e. the scattering layer, plays a critical role in
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balancing the competing process of light reflection at the top surface and the light diffraction for effective coupling to the slot-waveguide modes. To simplify the design problem associated with multiple governing factors, which often results in a complex solution space with numerous local optimums, the low-permittivity polymer material (ε = 2.89) was used for the scattering layer. In this case, using the effective medium approximation, the effective refractive index of the scattering layer is determined by averaging the dielectric and air regions, ranging from 1 to 2.89, which is much smaller than that of the cladding layer. Therefore, the reflection at the top surface due to the impedance mismatch (Cheng 1989) becomes insensitive to the variation of the scattering layer, while leaving the diffraction and the subsequent mode coupling being the dominating factors in the optimization process. A 32 × 32 coarse mesh is applied for this example, which is found to be adequate in resulting in efficient designs. Under the D4 symmetry constraint, the number of design variable N in this case is reduced to 136. Considering the coarse mesh and the simplified physics due to the choice of low-permittivity material in the scattering layer, no filtering is applied in this example. Starting from the initial generation composed of random designs, the GA based NGTOs converge to the optimal scattering patterns maximizing the absorption for incidence at λ = 400 nm, 500 nm and 600 nm, as shown in Fig. 3a, d and g, respectively, where the dashed square window denoting the unit cell. The optimized scattering structures converge to evident periodical patterns as one would expect for coupling of incident plane wave to the discrete photonic modes. Similar structures can be found in many of the light trapping structure design in the literature. The optimized scattering structures with periodic patterns imply the effective coupling of incident plane wave to the discrete photonic modes. This fact can also be verified using the Fourier analysis (Wang et al. 2013). Figure 3b, e, h, illustrate the Fourier transformations of the optimized scattering patterns. The circles representing the first two fundamental TM modes supported in the slot waveguide for each wavelength. The overlap between the modes and coupling channels provided by the scattering layer manifests the effective mode coupling. Moreover, the optimization convergence is achieved within 120 generations as shown in Fig. 3c, f and i. The significant enhancement on the light absorption through the optimization can be observed from Table 1, which summarizes the absorption coefficients of the initial designs and the optimized designs. It should be noted that the optimized absorption for λ = 500 nm is low compared with the other two incident wavelengths due to the deconstructive interference in this multi-layered solar cell system (Wang et al. 2013).
3.2.2 Light-trapping structure design using GA based NGTO for slot waveguide based solar cell with high-permittivity scattering material In this example, the dielectric material in the scattering layer is replaced with a high-permittivity dielectric (GaP) identical to the material used for the cladding layers. While the overall absorbing performance can be improved, the use of high-permittivity dielectric material in the scattering layer leads to a more complicated optimization problem with multiple governing factors in effect. In this case, the scattering layer with proper spatial filling ratio of the dielectric material, i.e. the design volume fraction, determines the reflection of the incident light due to the impedance mismatch and the simultaneous diffraction to the targeted slot-waveguide modes. Owning to the fact that the geometry change on the scattering layer has a significant influence on both diffraction and reflection, these two competing processes need to be addressed simultaneously during optimization. Competition between these two factors brings forth a large number of local optimums and the stronger scattering effect leads to severe physical resonances, which unfortunately increases the complexity of solution space. Moreover, stochastic search algorithms are vulnerable to “structural noise”, i.e. checkerboard patterns and redundant small scale features that appear during optimization iterations and in the final results (Wang and Tai 2005). They exhibit little contribution to absorption enhancement while dramatically increase the difficulty in device fabrication. These pitfalls in using high-permittivity scattering material are overcome by the proposed GA based NGTO approach. As a stochastic search algorithm, GA is robust to strong nonlinearity. Meanwhile, the discontinuous Heaviside density filter is applied to deal with checkerboard patterns and redundant small-scale features. In this test, a 64 × 64 mesh with filtering radius r = 3 is applied. Considering the symmetry constraint, the number of design variable N in this case equals 528. The optimized scattering patterns consisting 3x3 unit cells for incident wavelength of 400 nm, 500 nm and 600 nm, are shown in Fig. 4a, c and e, respectively. Compared with the case using the lowpermittivity scattering material, the more complicated physics leads to more complex geometries in the optimized designs to provide effective scattering. The absorption coefficients and enhancement factors of the optimized designs for each wavelength are summarized in Table 2. The light-trapping enhancement factor is defined as the ratio of the total absorption to the single-path absorption, and the Yablonovitch limit on the factor in for this structure is 7.22. All optimized designs exceed the light-trapping limit and achieve near-complete absorptions, ranging from 0.96 to 0.99, at the targeting wavelength
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without using a back reflector. Meanwhile, the enhancement factors (Wang et al. 2013) shown in Table 2 are far beyond the Yablonovitch limit, which shows that the effective light-trapping effects are attained through optimizations. The optimization results illustrate that both reflection and transmission are eliminated and the whole diffracted energy is absorbed by the active layer. In the spectral performance plots shown in Fig. 4 b, d and f, the distinctive absorption peaks at each target wavelength illustrate the effectiveness of optimization, whereas the numerous sharp peaks over the spectrum reveal the severe physical resonances for this design concept. Our investigation notes that different designs with similar light-trapping performance may be obtained from separate runs of GA based NGTO for a targeting incident wavelength. This fact confirms that the case of high-permittivity scattering material involves severe optimization nonlinearity and multiple local optimums. It should be noted that the nanophotonic light-trapping problem considered in this paper is within the linear optics regime, and the nonlinearity refers to the mul-
timodal design problem with complex solution space. This design nonlinearity is significantly increased for the case using the high-permittivity scattering material compared with the low-permittivity case at a single incident wavelength. Finally, an optimization of light trapping structure over the broad visible spectrum from 300 nm to 700 nm is performed. Designing proper scattering patterns for broadband performance becomes even more complicated than for a single wavelength. As slot waveguide supports discrete
Table 1 Absorption coefficients of optimized designs using GA based NGTO for the low-permittivity scattering material case at different incident wavelengths Targeting Wavelength (nm)
λ = 400
λ = 500
λ = 600
Initial Design Optimized Design
0.012 0.890
0.011 0.026
0.016 0.499
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resonances, the spectral absorption depends on contributions from the aggregated resonances. It is expected that the overall spectral absorption can be enhanced with the improvement of absorption of multiple wavelengths. Therefore, 31 wavelengths evenly distributed in the spectrum are considered in this broadband optimization. The optimized pattern and the spectral absorption from 300 nm to 700 nm are shown in Fig. 5a and e respectively. The average absorption over the whole spectrum reaches 0.481, with a light-trapping enhancement factor (Wang et al. 2013) of 24.4, which is above three times the Yablonovitch Limit at the normal incidence. For comparison, the spectral absorption of the random scattering layer using the same material is depicted in Fig. 5e. In contrast to both
Table 2 Absorption coefficient and enhancement factor of optimized design using GA based NGTO for the high-permittivity scattering material case at different incident wavelengths Targeting Wavelength (nm)
λ = 400
λ = 500
λ = 600
Absorption Coefficient Enhancement Factor
0.964 39.61
0.985 50.67
0.991 60.78
the strong coupling to slot waveguide modes and reduction of reflection from optimized scattering patterns, the random scattering pattern provides a continuum of wave vectors and results into a much lower absorption coefficient of 0.04 and an enhancement factor 2.03 inferior to the Yablonovitch Limit. This comparison demonstrates that without elaborate designs of the topology of scattering layer, slot-waveguide based cells cannot exhibit superior spectral performance. 3.3 Fabrication feasibility To demonstrate the fabrication feasibility of the optimized design, the broadband optimized structure in Fig. 5a is realized experimentally. It is observed that dot-shaped features at very small scale appear in the optimized designs using high-permittivity scattering material, as shown in Fig. 4a, c, d and Fig. 5a. This is mainly caused by using a threshold value of 0.5 during the filtering process. As noted in (Wang et al. 2011) that minimum length scale is not ensured by using a projection filtering with threshold value between 0 and 1. In future work, a filter with threshold at 0 and a shorter filtering radius will be adopted to prevent the loss of length scale control. Considering the insignificant contribution of these redundant
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(e)
(b)
3 µm
(c)
(d)
Optimized Design Fabricated Sample Random Pattern
Absorption Coefficient
(a)
500 nm
0.5
0 300
500
700
Wavelength Fig. 5 Broadband optimization results using GA based NGTO for the high-permittivity scattering material case and the fabrication example
small features, they are neglected during the fabrication process. The dashed square window stands for the unit cell of a 600 nm × 600 nm size. Considering the small features of a length scale around 20 nm, the electron beam lithography is used to fabricate this structure. After going through processes including spin-coating of photoresist, patterned with exposure to electron beam, etching and liftoff process, the fabricated pattern is shown in Fig. 5b and c with scale of 3 μm and 500 nm, respectively. The fabricated pattern agrees well with the optimized design on a large scale as shown in Fig. 5b. However, certain inevitable imperfections including the smoother boundary, the deformation or the absence of small features are observed at small scale shown in Fig. 5c. These imperfections can be alleviated after multiple experimental trials. For evaluating the performance of the fabricated pattern, the sample is transformed back into a bitmap graphic as shown in Fig. 5d. Using RCWA, the absorption coefficient of the fabricated structure over the spectrum is shown in Fig. 5e. The similar trend is observed in the performance curves of the optimized design and the fabricated structure with imperfection, whereas certain mismatch between the two curves are discovered at the locations where sharp peaks or dips happen. Since this solar cell is a highly nonlinear optical system, a minor perturbation in the scattering pattern will result in the shift of resonances. However, the broadband performance, which is the average absorption over the whole spectrum, will not be significantly affected by the resonance shifting. As shown in Fig. 5e, the average absorption of the fabricated structure is 0.472, a reduction less than 2 % of the 0.481 absorption coefficient of the optimized design. This further exhibits the robustness of our
broadband optimized design with respect to the fabrication imperfections.
4 SIMP based GTO versus GA based NGTO for light-trapping structure optimization 4.1 SIMP based GTO approach While the proposed GA based NGTO approach has been successfully applied to achieve highly efficient lighttrapping structures, the SIMP based method is also implemented in this study as an exploration of using GTO approach to solve this nonlinear problem. The SIMP method (Sigmund 1997) is currently the most popular GTO method. The original discrete value (0/1) problem is relaxed to a continuous variable optimization problem to enable the gradient calculation. The intermediate pseudo density value between 0 and 1 is penalized with a power law under a volume constraint. Using appropriate filtering techniques (Sigmund 2007), an almost 0/1 discrete design can be obtained. Recent years have seen successful applications of the SIMP method in designing high-performance nanophotonic devices (Andkjaer and Sigmund 2011; Sigmund and Hougaard 2008; Jensen and Sigmund 2005). In typical nanophotonic optimization problems (Bellido and Donoso 2007), such as the low-loss waveguide design, the discrete 0/1 structure with sharp material contrast between elements is favored. For these problems, the optimization converges to a discrete design even without penalty on intermediate-valued elements. Nevertheless for nanophotonic light-trapping structure designs in this work, our study shows that optimization sometimes converges to designs
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with a large amount of intermediate-valued elements without penalty. Such design is infeasible for the fabrication. To overcome this difficulty, a widely used approach in literature is the density filtering method with an approximated Heaviside function (Sigmund 2007; Guest et al. 2004). An almost 0/1 design is achieved by gradually increasing the curvature parameter β in the approximated Heaviside function using a continuation method. In this work, the modified Heaviside density filter (Guest et al. 2011) termed as the constant Heaviside density filter, is adopted. The β continuation is eliminated by increasing the upper bound of the design variables xi from 1 to xmax . The issue of intermediate-valued density can be effectively suppressed by ensuring a large magnitude of β·xmax . For forward analysis, the RCWA (see Section 2.3), the same analysis model used in the GA based NGTO is adopted. The gradient information is obtained by calculating the sensitivity of the absorption coefficient to the material property, i.e. the permittivity ε. The method proposed in (van der Aa and Mattheij 2007) to compute the gradient information based on the RCWA for 1D grating structure parametric design is extended in this work for 2D topology optimization of light-trapping structures. The analytical gradients computation involves calculating the sensitivities of eigenvalues and eigenvectors of the field governing eigen-equation in the scattering layer with respect to the permittivity ε. Next, the S matrix algorithm is again used for integrating the analysis across different layers. It should be noted that numerical inaccuracy may arise for finitedifference gradient evaluations when there exists a strong nonlinearity associated with resonance as to be shown in the comparison study in Section 4.2. With the gradient information, a design is updated using the method of Moving Asymptotes (MMA) optimizer [65]. As recommended in (Guest et al. 2011; Svanberg 1987), a conservative MMA with tightened asymptotes (compared with the default settings provided in (Svanberg 2004)) is adopted to prevent oscillations in optimization process considering the high nonlinearity. Considering the distinct governing physics for the slot-waveguide based thin-film solar cell with different scattering materials, both cases of low and high-permittivity scattering materials are discussed and solved using SIMP based method. 4.2 Nanophotonic light-trapping structure optimization results using SIMP based GTO Using SIMP based approach to design the nanophotonic light-trapping structure is explored in this section based on the solar cell model shown in Fig. 1c. It has been discovered so far that using the low-permittivity scattering material will result in a less nonlinear design problem, whereas the high-permittivity material in the scattering layer will lead
to a highly nonlinear problem with severe resonance and multiple local optimums. The performance of the SIMP based approach in handling problems with these two categories of behaviors will be discussed. This exploration will provide insights into applicability of SIMP based method to nanophotonic designs under different conditions and the opportunity for future work. 4.2.1 Light-trapping structure design using SIMP based GTO for slot waveguide based solar cell with low-permittivity scattering material In this test, the solar cell structure is shown in Fig. 1c with the low-permittivity scattering material. The unit cell is discretized using a 64 × 64 mesh without any symmetry constraint on geometry. Two incident polarization angles of π/4 and 3π/4 are simultaneously applied for illustrating the advantage of symmetric geometry in designing periodic light-trapping structures. The constant Heaviside density filter without β continuation is adopted with r = 3, β = 2∼4, and xmax = 3, where r denotes the filtering radius. The optimization starting from random designs and no volume constraint is applied. The optimized designs (3 × 3 cells array) for single incident wavelength of 400 nm, 500 nm and 600 nm using SIMP based GTO are shown in Fig. 6a, c and e, respectively. By comparing with Fig. 3b, e and h, it is observed that the SIMP based GTOs converge to the very similar designs as those from GA based NGTO. This significant similarity verifies the validity of using both approaches in delivering efficient nanophotonic light-trapping structure for low-permittivity scattering materials. Moreover, since no symmetry constraint is imposed in this case, the appearance of the symmetric geometry in optimized designs proves the superiority of symmetric geometry in periodic structure for light trapping under the natural condition of complete incident polarizations. Fig. 6b, d and f show the optimization histories of the achieved absorption coefficient value and the maximum magnitude of design change. The absorption coefficients of the optimized designs using the SIMP based approach are summarized in Table 3 with a comparison to those using GA based approach. It shows that the SIMP based GTO achieves close, albeit lower values of absorption coefficient in the optimized designs comparing to those from NGTO. As discussed in Section 3.2, the simplified physics in this case results in a solution space with less local optimums. Moreover, the mild scattering effect provided by the low-permittivity scattering material leads to limited resonance, thus a less nonlinear design problem. This is verified in our study using different starting designs and obtaining the same optimized patterns. It has been noted from comparison shown in Table 3 that the optimal performance from using SIMP is slightly
Author's personal copy Topology optimization for light-trapping structures Fig. 6 Optimization results using SIMP based GTO for the low-permittivity scattering material case at different wavelengths
inferior to the GA optimized results. This is attributed to the intermediate-valued elements of the optimized structure from using the SIMP method. The performances of most nanophotonic devices are sensitive to small structural perturbations. This issue can be mitigated to some extent by using a larger magnitude of β · xmax in the constant Heaviside density filter. However, to push the result to an almost 0/1 design, the SIMP power-law penalty along with a reasonable volume constraint is necessary. Therefore, a further investigation is conducted for applying a volume constraint. The SIMP penalty is set as 3, and the optimization for 600 nm-wavelength incidence is chosen as the testing case.
Table 3 Comparison of Absorption Coefficients of Optimized Designs using SIMP based GTO and using GA based NGTO for the low-permittivity scattering material case at different incident wavelengths Targeting Wavelength (nm)
λ = 400
λ = 500
λ = 600
SIMP GA
0.701 0.890
0.026 0.026
0.488 0.499
By regarding the unit cell shown in Fig. 3g as the optimal low-permittivity structure design at 600 nm incident wavelength, the volume faction in this case is around 63 %. Thus three different volume constrains, i.e. ≤60 %, ≤65 % and ≤70 %, are tested respectively. The optimization results are shown in Fig. 7a and b, c and d, and e and f respectively. As shown in Fig. 7a and c, the optimization converge to much more discrete designs due to the strict volume constraint, compared with the result in Fig. 6e where no volume constraint is applied. As shown by Fig. 7b and d, the volume constraints are kept tight at the latter stage of the optimization in these two cases, effectively pushing the element densities toward 0 or 1. Meanwhile, topological difference is observed between Fig. 7a and Fig. 3g, which is also attributed to the over-strict volume constraint of ≤60 %. As a comparison, the loose volume constraint at the latter stage of the optimization under the volume constraint of ≤70 % (Fig. 7f) explains the less discreteness in the optimized pattern in Fig. 7e. Despite the different topologies, optimized designs in Fig. 7(a), (c) and (e) achieve similar absorption as 0.475, 0.496 and 0.491, respectively. Fig. 7c performs the best considering that it is closer to the optimum. These observations indicate that the optimal value for the volume constraint lies around 65 %.
Author's personal copy S. Yu et al. Fig. 7 Optimization results using SIMP based GTO for the low-permittivity scattering material case at 600 nm incident wavelength with different volume constraints
Therefore, for the case of low-permittivity scattering material, SIMP based approach can achieve efficient lighttrapping structure. An almost 0/1 optimized design can be obtained with an appropriate volume constraint and the Heaviside filtering. Applying a volume constraint may affect the final topology and the performance of the optimized structure. Considering the efficiency of the GTO approach, the appropriate value for the volume constraint needs to be determined through a trial-and-error process. 4.2.2 Light-trapping structure design using SIMP based GTO for slot waveguide based solar cell with high-permittivity scattering material In this test, the high-permittivity material is used for the scattering layer of the slot-waveguide based cell. While there is a potential to achieve a complete absorption as shown in Fig. 4 and Table 2, the strong scattering effect and the complex physics of competing governing processes result in a large number of local optimums in the solution
space. This is confirmed in our study that separate runs of GA based NGTO for high-permittivity material case may converge to designs achieving nearly complete absorption at the targeting wavelength but with differing topologies. This issue poses a great difficulty to GTO approaches. Based on our study, it is challenging for the direct implementation of SIMP based method to capture the local resonance at a single incident wavelength as the peak shown in Fig. 4b, d and f. Practically, the thin-film cell performance is affected by a broadband light absorption over the solar spectrum. Fortunately, the severe nonlinearity caused by the nanophotonic resonance is reduced when multiple wavelengths are simultaneously considered in a broadband design problem. As a demonstration, the light-trapping structure made of highpermittivity scattering material is optimized using SIMP based approach considering three incident wavelengths, i.e. 400 nm, 500 nm, and 600 nm. For this optimization, the SIMP penalization factor is set as 3, the constant Heaviside filter with β · xmax at 24 is adopted, and a 40 % volume constraint is applied. The optimization results are
Author's personal copy Topology optimization for light-trapping structures Fig. 8 Optimization results using SIMP based GTO for high-permittivity scattering material case at multiple wavelengths
shown in Fig. 8 here 8 a, b and c. An averaged 0.52 absorption over the three targeting wavelengths is obtained. The intermediate-valued elements are observed in optimized pattern, and the volume constraint is loose at the latter stage of optimization as shown in Fig. 7b. As mentioned before, different volume constraints may result into optimized designs with different topologies. In some cases such as the example shown in Fig. 7a and b, the converged designs have lower volume fraction values than the constraints applied. Under such situation, increasing the magnitude of β ·xmax or adopting the Heaviside filtering with β continuation (Wang et al. 2011) may mitigate the issue of intermediate-valued elements. To obtain a broadband efficient light-trapping design, 31 wavelengths are evenly selected from 300 nm to 700 nm for the optimization. Same settings as for the case optimizing 3 wavelengths are adopted, except a ≤50 % volume
constraint is applied. A broadband efficient light-trapping structure is obtained as shown in Fig. 8c. Based on the optimization history shown in Fig. 8d, an averaged absorption of 0.435 is achieved over the 31 selected wavelengths. The ≤50 % volume constraint is satisfied and kept tight at the latter stage of the optimization, which leads to an almost 0/1 design. The performance of the optimized design is further validated over the whole spectrum from 300 nm to 700 nm. As shown in Fig. 8e, an averaged absorption of 0.420 is achieved for the solar spectrum. The difference between the absorption over the whole spectrum and the 31 targeting wavelengths is attributed to the fact that not all incident wavelengths from 300 nm to 700 nm are considered for the optimization. The different selection of incident wavelengths for the broadband optimization in this work may lead to a different design with varying absorbing performance. Compared with the optimized results from GA
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based NGTO shown in Fig. 5a and e, comparable absorbing performance is achieved using SIMP based approach. Meanwhile, more discrete pattern can be obtained by adjusting the volume constraint and the Heaviside filtering. It is noted that the SIMP provides better length-scale control than GA in this work for the multi-wavelength optimization using high-permittivity scattering material. In our work, it takes approximately 90 CPU hours for convergence of the broadband GA based NGTO implemented in 12 paralleled threads on a multi-core desktop. Although the adjoint method for sensitivity analysis is not exploited in this work, the broadband GTO is fivetime faster than the broadband NGTO in achieving optimized structure with comparable performance. Utilizing the adjoint method in the future work will further increase the computational efficiency of GTO significantly, which will provide this approach with more freedom in considering other factors during optimization. For example, more incident wavelengths may be considered for the broadband optimization, or the impact of fabrication imperfection may be dealt with during the optimization process using robust design formulation.
5 Conclusion This paper presents a general, yet systematic design approach based on topology optimization for designing highly efficient nanophotonic light-trapping structure in solar cell beyond the reach of conventional intuitive designs. By exploiting the power of topology optimization, our approach searches for the optimal structure layout of the scattering layer in solar cell, and the optimized designs are superior to existing solar cell designs that focus on either layer thicknesses or material composition. Our approach is generally applicable to various solar cell devices with different light-trapping schemes and can be readily extended to other nanophotonic designs. The approach based on GA has shown to be robust in dealing with the strong nonlinearity involved in the nanophotonic light-trapping design problem. RCWA, which is efficient for analyzing scattering in multi-layered systems with periodic structures, is introduced into the TO framework (either for GA or SIMP) to facilitate the simultaneous considerations of all governing physics over the optical absorption in solar cell devices. Through the implementation of both the GA based NGTO approach and the SIMP based GTO approach to design the light-trapping structure using different material, insights into both approaches in dealing with nanophotonic design problems are provided along with the opportunities for future work. Solar cell models with the corresponding light-trapping process governed by differing physics are tested, including
the slot-waveguide based cell with low-permittivity scattering material and high-permittivity scattering material. The use of low-permittivity scattering material simplifies the physics in nanophotonic light-trapping process and reduces the nonlinearity of the design problem. The highpermittivity material in the scattering layer of the cell leads to a highly nonlinear problem, which poses a challenge to the design approach. In this case, the introduction of the discontinuous Heaviside density filter effectively suppresses the numerical instability involved. In spite of the distinct physical characteristics, highly efficient light-trapping structures are obtained for these cases using the proposed approach. The optimized designs are shown to achieve significant enhancements in light absorption, exceeding the Yablonovitch limit on light-trapping effect. This approach can be readily applied to various solar cell models with different light-trapping schemes. We have optimized the light-trapping structures of the thin-film silicon solar cell and the plasmonic-based thin-film cell using the proposed approach; these results are not included in this paper. We have successfully fabricated the broadband optimized pattern using the electron beam lithography. The fabricated structure matches well with the optimal design in spite of some fabrication imperfections. Based on the analysis using RCWA, it is found that there is very limited difference in the light-trapping performance between the fabricated sample and the optimized design over the broad spectrum. In future work, the complete cell device will be fabricated and the actual efficiency in energy conversion will be tested. This optimized organic solar cell design requires much less materials, hence lower manufacturing cost, while still maintaining the efficiency. There is a great potential of using this solar cell as a cost-effective clean energy source. For the low-permittivity scattering material case with less nonlinearity, similar optimized design as those from GA based NGTO are obtained using SIMP based GTO, which verifies both approaches. By applying an appropriate volume constraint and filtering technique in SIMP based method, almost 0/1 discrete optimized patterns can be obtained. For the case using high-permittivity scattering material, the complicated physical process and the strong scattering effect lead to a complex solution space with multiple optimums and severe nonlinearity. It is challenging for the SIMP based approach to capture the local resonance in solving single incident wavelength optimization. However, for the broadband optimization with simultaneous consideration of multiple wavelengths, efficient designs using high-permittivity material can be obtained with SIMP based approach at a much lower computational efforts compared to NGTO approaches. GA based method requires millions of function evaluation using the RCWA analysis code to solve the problem due to its non-gradient characteristic; for SIMP based method, the number of function evaluation will
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be reduced to two for each iteration by exploiting the adjoint method for efficient sensitivity analysis. Such computational efficiency provides GTO approaches with advantage over NGTO approaches in considering more wavelengths and other factors, such as the geometric uncertainty due to fabrication imperfections, during optimization. Moreover, for other nanophotonic design problems with a larger number of design variables or without geometric symmetry [28], NGTO may fail to converge to the optimum due to the curse of design dimensionality. It is also noted in this work that although the SIMP based GTO requires careful dealing with the issue of intermediated-valued elements, this approach provides better length scale control than the GA based NGTO in solving the high-permittivity scattering material case. In future work, for using the SIMP based GTO to optimize the light-trapping structure, efficient analytical sensitivity evaluation based on adjoint method needs to be developed. Acknowledgments This work is supported by the National Science Foundation under Grant number CMMI-1130640 and CMMI0751621. Shuangcheng Yu would like to thank the International Institute for Nanotechnology for the Ryan Fellowship award. Dr. Fan Zhou at Northwestern University is acknowledged for the fabrication work. The authors would like to thank Prof. Krister Svanberg at KTH Royal Institute of Technology for providing the MMA code. The authors would also thank the reviewers for helpful comments that improved this manuscript.
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