# Design and Analysis of Spectrally Selective Patterned Thin-Film Cells

## Authors

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DOI: 10.1007/s10765-013-1495-y

- Cite this article as:
- Hajimirza, S. & Howell, J.R. Int J Thermophys (2013) 34: 1930. doi:10.1007/s10765-013-1495-y

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## Abstract

This paper outlines several techniques for systematic and efficient optimization as well as sensitivity assessment to fabrication tolerances of surface texturing patterns in thin film amorphous silicon (a-Si) solar cells. The aim is to achieve maximum absorption enhancement. The joint optimization of several geometrical parameters of a three-dimensional lattice of periodic square silver nanoparticles, and an absorbing thin layer of a-Si, using constrained optimization tools and numerical FDTD simulations is reported. Global and local optimization methods, such as the Broyden–Fletcher–Goldfarb–Shanno quasi-Newton method and simulated annealing, are employed concurrently for solving the inverse near-field radiation problem. The design of the silver-patterned solar panel is optimized to yield maximum average enhancement in photon absorption over the solar spectrum. The optimization techniques are expedited and improved using a novel nonuniform adaptive spectral sampling technique. Furthermore, the sensitivity of the optimally designed parameters of the solar structure is analyzed by postulating a probabilistic model for the errors introduced in the fabrication process. Monte Carlo simulations and unscented transform techniques are used for this purpose.

### Keywords

Fabrication errorInverse optimizationSensitivity analysis Thin-film solar cells### List of Symbols

### Variables

- \(a\)
Variable in Eq. 7

- \(E\)
Enhancement factor

- \(f\)
Objective function

- \(h_{\mathrm{Ag}}\)
Height of silver nanowires

- \(h_{\mathrm{Si}}\)
Height of amorphous silicon

- \(H_n\)
Hermite polynomial of order \(N\)

- \(I\)
Solar irradiance spectrum

- \(S_i\)
Sigma points for the unscented transform

- \(T\)
Number of satisfied moments in unscented transform

- \(w_i\)
Sigma weights for the unscented transform

- \(w_{\mathrm{Ag}}\)
Width of silver nanowires

- \({\varvec{x}}\)
Selected geometry

- \(\Vert {\varvec{x}} \Vert \)
Norm of vector \({\varvec{x}}\)

### Greek Symbols

- \(\alpha _{\mathrm{gr}}\)
Spectral absorptivity in the presence of grating

- \(\alpha _{\mathrm{ngr}}\)
Spectral absorptivity in the absence of grating

- \(\lambda \)
Wavelength

- \(\gamma _i, \lambda _i\)
Predefined constants

- \(\Delta {\varvec{x}}\)
Change in \({\varvec{x}}\)

- \(\varLambda _{\mathrm{Ag}}\)
Nanowires period

- \(\varOmega \)
Optical wavelength range

## 1 Introduction

Today, fossil fuels constitute the largest fraction of global energy use. Looking at the historical trend of energy consumption, however, it is ostensible that the dominance of fossil fuel usage has declined significantly over the past 50 years [1], mostly due to limited natural resources, undesirable environmental effects, uneven geographical distribution, and political influences. In contrast, other forms of energy such as nuclear and renewable sources have gained higher usage in the global energy profile. In particular, solar photovoltaic (PV) energy has experienced the highest growth rate in the past decade, exemplified by a 102 % increase in utility scale usage over the last five years [2]. Whether or not PV energy will keep growing at such a fast pace depends on technological advances in efficient and inexpensive conversion of solar energy to electricity. To that aim, much research has been dedicated to increase the performance and reduce the cost of PV cells, so that they become competitive with current modes of electricity generation.

Most of the cost of a PV cell is in the material and processing expenses. At present, most commercial PV cells are based on bulk or wafer-based crystalline silicon (c-Si) which, although very efficient in transforming insolation to electricity, is quite expensive due to the excessive usage of semiconductor materials. An alternative to first generation c-Si PV cells is the use of thin-film semiconductor layers of non-crystalline silicon which requires anywhere between 10 and 300 times less material than traditional solar cells. In thin-film cells, a thin layer of a semiconductor is deposited on a suitable substrate using low cost techniques. Popular thin-film materials are amorphous silicon (a-Si), polycrystalline (p-Si) and nanocrystalline (nc-Si), cadmium telluride, copper indium selenide, and others. Despite the lower absorption and overall efficiency of silicon, it is widely used in thin-film solar cells due to its abundance, renewability, and the availability of a mature manufacturing process which is the consequence of many years of industrial experience in digital electronics.

Despite offering processing and cost advantages, the efficiency of the state-of-the-art thin-film PV cells (around 10 %) is noticeably less than (about half) that of c-Si PV cells, mainly due to the smaller thickness of the absorbing layers. Therefore, major efforts are focused on producing thin-film cells with higher efficiencies. A class of techniques employed to enhance the optical absorption of thin-film solar cells is called “light trapping” techniques, which refer to mechanisms used to increase the average path length of incident light inside the solar cell, and thereby improving the overall fraction of absorbed photons. A systematic design of light trapping mechanisms often entails analyzing radiative near-field effects to develop methods for enhancing both spectral and directional selectivity of solar cells. This is often achieved through nanopatterning of the PV surface.

Light trapping in thin-film solar cells via surface texturing or nanopatterning has been discussed in many prior publications. Four common techniques include the deposition of plasmonic nanoparticles on the front or back surface of the solar cells (see, e.g., [3–9]), surface texturing through the use of metallic gratings such as nanostrips or nanochips (see, e.g., [10–15]), textured transparent conductive oxide (TCO) [15–17], and the use of semiconductor nanowires [18–21]. Some of these references report a significant increase in the photonic absorption capability of the thin-film structure through the use of one or more of the light trapping techniques, as well as the right choice of material. In particular, Rockstuhl et al. [11] demonstrated that applying silver nanowires across the upper surface of a solar panel composed of thin-film a-Si can increase total photon absorption by as much as 60 %. Tumbleston et al. [12] claimed an increase in photon absorption of around 18 % in organic-based solar cells with photonic crystal structures and zinc oxide gratings. Beck et al. [13] reported increased optical absorption by a factor of five at a wavelength of 1100 nm, and enhanced external quantum efficiency of thin-film Si solar cells by a factor of 2.3 at the same wavelength via tuning localized surface plasmons in arrays of silver nanoparticles. Wang et al. [14] obtained 30 % broadband absorption enhancement for thin a-Si using unique nanogratings. Other aforementioned references have reported similar or even more promising results through exploiting combinations of light trapping techniques and advanced deposition methods.

In the present work, we focus on light trapping through the use of metallic nanogratings. The mechanisms responsible for the increase in absorption in this case include Fabry–Perot resonance, plasmon polariton generation on the surface, and resonance and planar waveguide coupling. The exact analysis of the light trapping properties of a given nanopattern is a formidable analytical/numerical problem, requiring the solution of near-field radiation-surface interactions through Maxwell’s equations for electromagnetic waves. Moreover, designing a nanopatterned surface with properties tailored for solar cells is even more difficult, as it represents an inverse design problem. This requires an inverse solution of Maxwell’s equations applied to interaction of electromagnetic waves with a surface geometry that is to be determined, given the desired spectral/directional distribution of absorbed radiation. Nevertheless, despite the scope and importance of the problem, scant work exists for dealing with the inverse solution of near-field radiation for nanopatterning of solar cells. Most work in the literature deals only with simple geometries using simplistic inverse methods for optimizing surface patterning (or in many cases, repetitive “brute-force” solutions covering the entire map of patterning parameters). Often, only one or two parameters are considered.

In previous work, we have reported the results of preliminary experiments of the inverse optimization method for certain classes of 2-dimensional and 3-dimensional solar cell structures [22, 23]. In the current work, we improve upon the numerical techniques of [22, 23] to incorporate more sophisticated and practical scenarios. We first optimize the surface patterning of 3-dimensional solar panels in the presence of a periodic lattice of square nanochip grating, in order to maximize the light absorption. In doing so, we provide a mathematical framework for the optimization program which facilitates the incorporation of various practical and physical constraints. We demonstrate that a hybrid numerical optimization (composed of simulated annealing followed by quasi-Newton optimization) finds a near-optimal solution within a limited number of iterations (far less than that required for an exhaustive search method). In particular, we find a geometry pattern in which the light trapping factor (i.e., number of absorbed photons) increases roughly by a factor of 1.52 when surface texturing is used. In addition to the optimization of 3-dimensional surface patterning, we introduce a novel approach for the approximation of spectral optical characteristics of a given cell, which results in faster convergence rates for absorption calculation. The proposed method is based on adaptive nonuniform sampling of a target irradiance-absorption spectrum, motivated by the inhomogeneous concentration of absorption spectrum and convergence rates of FDTD simulations in the optical range.^{1} We use the proposed method in the inverse optimization and report expedited simulations.

Finally, we evaluate the sensitivity of the inverse optimization solution in the presence of fabrication/modeling error. Employing a Monte Carlo (MC) technique, and a multidimensional unscented transform (UT) method, we statistically analyze the robustness of the proposed design in the presence of Gaussian error in all geometry parameters. A practical sensitivity analysis is a critical issue in the engineering of high efficiency solar cells, especially from a manufacturing point of view. Despite this importance, there is no prior work on this subject, and to the best of the authors’ knowledge, the present work is a first attempt at this problem.

## 2 Problem Setup

## 3 Optimization of the Geometry Parameters

The *quasi-Newton* (QN) method and *simulated annealing* (SA) are used to find the geometry corresponding to the maximum enhancement factor (EF). Both methods are detailed in prior work [22]. However, we provide a brief description of the two methods for completeness.

### 3.1 Quasi-Newton Method

The QN method is a memory-less optimization technique that is suited for finding local optima of a sufficiently smooth function. It consists of continuously updating a search point based on the first and second derivatives of the objective function [24–26]. A new candidate point is selected in the direction of the maximum descent determined by the gradient vector multiplied by the inverse of the Hessian matrix. The search for a new point with a better objective function proceeds in that direction, using a so-called line-search process. For enhancement factor maximization, the unconstrained objective (cost) function \(f({\varvec{x}})\) is taken as the reciprocal of the solar enhancement factor \(E({\varvec{x}})\), where \({{\varvec{x}}}\) is the vector of geometric parameters being optimized. The Broyden–Fletcher–Goldfarb–Shanno (BFGS) update for the Hessian matrix and the basic line-halving technique for the line search are chosen. For a more detailed description of the method, we refer the reader to [22, 23].

The QN method is deterministic in nature and may become trapped in local minima. Adding a perturbation in the search direction can improve its performance. However, it remains unlikely that this approach modifies the search region significantly, and therefore the algorithm might settle down at a local minimum far from the global solution. The QN method is therefore well-suited for situations where the initial candidate point happens to be close to the global minimum, or for optimization problems having a single optimum. Unfortunately, that is not the case here, as the electromagnetic objective function is highly nonlinear and has many local optima [22]. We will describe another optimization method which is more suitable for finding global solutions in such cases.

### 3.2 Simulated Annealing

In contrast to the QN method, the SA method is randomized in nature and is more suited for situations where the initial geometry is not known to reside in the proximity of the global minimum, and for highly non-convex objective functions [27, 28]. Although SA is primarily used for discrete optimization, variations have been introduced for solving continuous optimization problems. In this work, the fast annealing method developed by Ingber is chosen, in which a Cauchy distribution for determining new candidate points is used rather than the conventional Boltzmann distribution [29]. A detailed description of the SA method used is given in [22].

### 3.3 Hybrid Optimization Technique

Although SA is suitable for quickly scanning large state spaces, it often fails to capture some local variations of the objective function, especially when the cooling rate parameter is large. The performance of the SA method can be improved via techniques such as iterative re-annealing and cooling, use of memory, or alternative choices of the selection rule (see, for example, [30]). Also, we refer the interested reader to [31], for a discussion on the generalized SA method and how its performance can be tailored to different applications.

Another technique that can be used to improve the performance of the SA algorithm is to combine it with a local optimization technique to form a hybrid platform. Examples of local optimizers are the memory-less algorithms such as the described QN method, or the memory utilizing techniques such as Tabu Search (TS, see, e.g., [23] for optimization of 3-dimensional solar cell patterns using TS). In contrast to SA, the QN algorithm is more promising for the cases where the initial geometry is close to the global optimum. Hence, SA is more recognized as a global solver, whereas QN is a local solver. It is in general useful to exploit a combination of the two methods for a fast and accurate global optimization (see, e.g., [32–34] and the references therein for a more sophisticated study of hybrid optimization tools based on SA). In this paper, we evaluate the performance of the SA and QN optimization techniques individually, as well as in a hybrid framework to optimize the three-dimensional parameters of a surface-textured a-Si solar panel. In the hybrid optimization, a fast converging SA is run to achieve a set of one or more candidate point(s) with suboptimal objective functions, including the best solution found by SA. Afterwards, the QN method is implemented starting from the set of candidate points to perform a proximity search. The numerical results and the discussions of the implemented optimization techniques are given in Sect. 7.

## 4 Practical Constraints

### 4.1 Constrained Optimization

As discussed above, there are several sources of constraints that are enforced on the set of valid geometries. The first type of constraint includes upper and lower bounds on each dimension. These bounds are chosen so that nanochip sizes and the thin-film thickness are not unreasonably large or unrealistically small. Provided that the underlying electromagnetic interaction of the incident light and the thin-film panel is relatively well understood, and the dependency of the absorbed light spectrum to the geometry parameters is (at least intuitively) known, it is possible to narrow down the space of possible dimensional values (“search space”) where using the nanochips can result in significant enhancement.

A second class of constraints involves those that arise from the available finite precision of the fabrication and testing devices. We must assume a limited precision for the allowable geometry parameters, since it is impossible to fabricate nanostructures with infinitesimally small precision. In this paper, due to limitations in computation time using the FDTD simulator, we limit the search space to parameters rounded off to \(1\,\hbox {nm}\), meaning that none of the dimensions have sub-nanometer numerical values.

*Lagrange multipliers*in the theory of constrained optimization. In fact, provided that the right \(\lambda _i \) and \(\gamma _i \) are chosen, from the theory of constrained optimization and Karush–Kuhn–Tucker (KKT) conditions [35], the solution to the original problem of Eq. 4 is given by the following unconstrained problem:

In contrast, in the SA algorithm, the choice of the objective function is not changed. Instead, the constraints are strictly imposed at every iteration by assuring that the new candidate point lies inside the allowable region. In other words, the rule for selection of a new candidate changes as follows. In each iteration, the selection of a new random candidate is based on the default distribution. However, if the new point falls outside the allowable region, the candidate is immediately rejected, and a new choice is considered. This is repeated until a valid candidate is selected.

## 5 Nonuniform Spectral Sampling and Fast Numerical Method

As expressed in Eq. 3, calculating the enhancement factor requires computing numerical integrals of the irradiance-absorption products in the absence and presence of a nanograting. FDTD simulations are set to compute the absorption power as a function of wavelength. Once the absorption power is computed for a sufficiently large number of wavelengths in the solar range, the irradiance-absorption product can be interpolated for the whole range and numerically integrated. The straightforward approach is to consider uniform wavelength samples in the solar range. Clearly, the more samples that are considered, the more accurate is the estimate of the enhancement factor. This is the standard approach that we have pursued for most of the numerical results of the current paper and in our previous work [22]. Certain aspects of the numerical computations reveal that a nonuniform choice of sample wavelengths results in more accurate estimates of the enhancement factor and hence a faster computational process. The key is the fact that FDTD simulations have variable convergence rates that depend on the physical nature of the problem and wavelength of simulations. As a result, the simulations tend to run longer at certain frequencies. Furthermore, in all of the empirical simulations, the FDTD simulations are much more time consuming in the presence of a grating. In addition, the irradiance-absorption profile is often a bandpass spectrum which is significantly larger in certain segments of the solar range. Therefore, it might be possible to reduce the computational costs of the simulations by employing a nonuniform set of frequency samples that are more concentrated at the bulk of the irradiance-absorption profile. We propose a simple nonuniform sampling approach that realizes this idea. The method is as follows.

## 6 Fabrication Error Modeling

Optimal design of periodic nanochips for achieving broadband absorption enhancement must be robust to incurred structural and numerical errors. In particular, every fabrication method is subject to errors that can be statistically modeled by examining a large sample set of finished structures based on a given target geometry, by means of high resolution nanoscale imaging techniques such as scanning electron microscopy (SEM) or atomic force microscopy (AFM). There are many factors that contribute to the fabrication error, including limited precision of fabrication devices/processes and human factors. To the best of our knowledge, a comprehensive study of nanostructure fabrication error modeling does not exist in the literature. In addition to concerns over errors that arise in fabrication reasons, it is also useful to know from a computational perspective how sensitive the optimal solution is. In particular, if the enhancement factor objective function is a highly oscillating function with respect to the geometry vector, then it is likely that good solutions are in the close vicinity of poor solutions, and therefore a small deviation in the structure can result in unexpectedly bad performance.

For simplicity, we consider a hypothetical error model in which in the finished cell, each dimension has an independent Gaussian deviation around the optimized target value, with some fixed standard deviation. In practice, structural errors of different parameters can be correlated, and their variances might depend on their values. However, in the absence of information on such possible correlations, we believe that the uncorrelated model is general enough to capture the effects of many uncertainties, and can be useful for evaluating the sensitivity of a proposed design.

## 7 Simulations and Discussion

### 7.1 Optimization Results

Results for SA and QN algorithms for the 3-D inverse optimization problem

Method | \({\varvec{x}}_{\mathrm{initial}} \)(nm) | Optimal EF | # of Trials | \({\varvec{x}}_{\mathrm{opt}} \)(nm) |
---|---|---|---|---|

SA1 | [129,145,53,230] | 1.32 | 12 | \(\left[ {81, 64, 76, 230} \right] \) |

SA1-QN | [129,145,53,230] | 1.39 | 24 | \([78, 64, 76, 230]\) |

QN1 | [125,54,70,222] | 1.274 | 31 | [60,62,61,252] |

SA2 | [79,85,69,202] | 1.51 | 15 | \([78, 62, 65, 183]\) |

SA2-QN | [79,85,69,202] | 1.527 | 19 | \([78, 62, 65, 184]\) |

QN2 | [60,62,67,182] | 1.243 | 30 | [128,73,81,227] |

### 7.2 Fabrication Error Modeling

Four optimum UT Sigma points and weights for Gaussian error in one dimension with standard deviation of 5 nm

Sigma points (\({\varvec{s}}_{\varvec{i}}, 1\le i\le 4)\) | \(\pm \)0.7420 | \(\pm \) 2.3344 |

Weights (\({\varvec{w}}_{\varvec{i}}, 1\le i\le 4)\) | 0.4541 | 0.0459 |

Ten optimum UT Sigma points and weights for Gaussian error in one dimension with standard deviation of 5 nm

\({\varvec{s}}_{\varvec{i}}\) | \(\pm \)2.4247 | \(\pm \)7.3299 | \(\pm \)12.4216 | \(\pm \)17.9091 | \(\pm \)24.2973 |

\({\varvec{w}}_{\varvec{i}}\) | 0.3446423 | 0.135484 | 0.019112 | 0.000758 | 0.000004 |

Mean, standard deviation, skewness, and kurtosis of the enhancement factor for Gaussian error with 5 nm standard deviation in each variable, estimated through UT with 10 sigma points

Variable parameter | Mean | Std. deviation | Skewness | Kurtosis |
---|---|---|---|---|

\(h_{\mathrm{Si}} \) | 1.441 | 0.05 | \(-\)0.3233 | 2.5365 |

\(h_{\mathrm{Ag}} \) | 1.500 | 0.04 | \(-\)0.4864 | 2.6248 |

\(w_{\mathrm{Ag}} /2\) | 1.405 | 0.05 | \(-\)0.0993 | 4.1824 |

\(\varLambda _{\mathrm{Ag}} \) | 1.393 | 0.06 | \(-\)0.2961 | 1.9539 |

### 7.3 Nonuniform Sampling Results

## 8 Conclusion

We studied the problem of inverse optimization in thin-film solar cells, for the optimal design of surface nanopatterns. We invoked mathematical tools of constrained optimization to formulate an optimization program to solve for the dimensions of a 3D periodic surface pattern maximizing the solar absorption enhancement factor. Using a constrained SA optimization followed by the localized QN method, we obtained an enhancement factor of 1.52 in the absorption of the solar power when silver nanopatterns are used. Furthermore, we proposed an adaptive sampling scheme that expedites the running time of spectral FDTD simulations. The proposed method is useful beyond this particular optimization problem and potentially reaches many other instances of electromagnetic profile optimization.

The suggested design obtained by the inverse optimization program is relatively resilient to fabrication error; in the presence of a Gaussian error in each geometry dimension with a 5 nm standard deviation, we demonstrated by using MC and unscented transform simulations that with 90 % confidence, the final enhancement factor is above 1.3.

Future work includes rigorous analysis of the convergence time of the proposed adaptive algorithm, as well as incorporation of more realistic fabrication error models. The inverse optimization paradigm introduced in this paper is a powerful tool that can be applied to other nanostructures including higher efficiency tandem thin-film cells, and can accommodate other physical constraints such as carrier recombination. These shall be the subjects of future research.

The term irradiance-absorption profile, and the details of the mentioned method shall be formally outlined in the remainder of the paper.

## Acknowledgments

The authors appreciate support for this work from the US National Science Foundation under Grant CBET-1032415 and also would like to thank Dr. Alex Heltzel for helpful discussions.