Quasi-random nanostructures are playing an increasingly important role in developing advanced material systems with various functionalities. Current development of functional quasi-random nanostructured material systems (NMSs) mainly follows a sequential strategy without considering the fabrication conditions in nanostructure optimization, which limits the feasibility of the optimized design for large-scale, parallel nanomanufacturing using bottom-up processes. We propose a novel design methodology for designing isotropic quasi-random NMSs that employs spectral density function (SDF) to concurrently optimize the nanostructure and design the corresponding nanomanufacturing conditions of a bottom-up process. Alternative to the well-known correlation functions for characterizing the structural correlation of NMSs, the SDF provides a convenient and informative design representation that maps processing–structure relation to enable fast explorations of optimal fabricable nanostructures and to exploit the stochastic nature of manufacturing processes. In this paper, we first introduce the SDF as a nondeterministic design representation for quasi-random NMSs, as an alternative to the two-point correlation function. Efficient reconstruction methods for quasi-random NMSs are developed for handling different morphologies, such as the channel-type and particle-type, in simulation-based microstructural design. The SDF-based computational design methodology is illustrated by the optimization of quasi-random light-trapping nanostructures in thin-film solar cells for both channel-type and particle-type NMSs. Finally, the concurrent design strategy is employed to optimize the quasi-random light-trapping structure manufactured via scalable wrinkle nanolithography process.

## Introduction

Functional nanostructured material systems (NMSs) fabricated using nanomanufacturing technologies have opened up new possibilities in many engineering applications. One category of NMSs, as represented by the metamaterial systems [13], usually relies on the periodic arrangement of identical building blocks to achieve the desired functionalities. For example, Fig. 1 shows a periodic NMS design (Fig. 1(a)) and the fabricated sample (Fig. 1(b)) of nanophotonic light-trapping structures to be utilized in thin-film solar cells. Designed using the topology optimization approach [5,6], the periodic structure exhibits a significant enhancement in light absorption compared with random pattern structures as shown in Fig. 1(c) [4]. However, such periodic NMS with carefully designed building blocks usually requires expensive and time-consuming top-down nanomanufacturing [7]. Top-down nanomanufacturing refers to those processing methods that reduce large pieces of materials all the way down to the nanoscale, like carving a model airplane out of a block of wood [8]. Examples of top-down nanomanufacturing include focused ion-beam milling, photolithography, and electron-beam lithography. Patterning the optimized design in Fig. 1(a) onto a 4 in wafer using electron-beam lithography normally takes more than a week. Such high manufacturing cost impedes the adoption of functional NMS at industrial scale.

Fig. 1
Fig. 1
Close modal

In this paper, we present a new design approach for designing cost-effective functional NMSs with quasi-random structures. In contrast to periodic structures, quasi-random NMSs are naturally formed via cost-effective, scalable bottom-up processes. Bottom-up nanomanufacturing creates nanostructures by building them up from atomic- and molecular-scale components [9], typical processes of which include the phase separation of polymer mixtures [10] and the mechanical self-assembly based on thin-film wrinkling [11]. The NMSs obtained via such processes usually comprise no periodic building blocks but a nondeterministic material distribution governed by underlying structural correlation. The structural correlation, which is determined by the physics of a bottom-up process, largely affects the macroscopic performance of quasi-random NMSs.

Multiple lineages of birds and insects have been discovered utilizing the quasi-random nanostructures in their feathers or scales to produce angle-independent structural colors [11,12]. As illustrated in Fig. 2, the longhorn beetle Anoplophora graafi in Fig. 2(a) possesses the sphere-type quasi-random NMS to produce the structural color of the light stripes on dark scales [13]; the beetle Sphingnotus mirabilis in Fig. 2(b) displays the white colored stripes produced by the channel-type quasi-random NMS in its scale [14]. Despite their different morphologies, these biological quasi-random NMSs are self-assembled via the phase separation of keratin and air [11,15]. The interplay between entropy and molecular interaction during the bottom-up structure formation process results in a pronounced short-range correlation but little long-range order, leading to the inherent quasi-randomness of the NMS.

Fig. 2
Fig. 2
Close modal

Man-made quasi-random NMSs have been developed using bottom-up nanomanufacturing. For instances, Fig. 2(c) shows a noniridescent structural color coating (shown by the inset) with sphere-type quasi-random nanostructure obtained from the self-assembly of polymer and cuttlefish ink nanoparticles [16]; Fig. 2(d) shows a hierarchical NMS with channel-type quasi-random nanostructure fabricated via nanowrinkling process [17], which possesses the superhydrophobic property as shown by the inset [18]. In spite of realization of a wide variety of man-made quasi-random NMSs, there is a lack of systematic design methodology for developing nondeterministic, nonperiodic nanostructures from bottom-up nanomanufacturing. Current development of functional quasi-random NMSs mainly follows a sequential strategy [19]. The optimal structure is first identified using computational design methods, and then the fabrication process is conceived and implemented to realize the optimal design. Without considering the fabrication conditions in structure optimization limits the feasibility of an optimized design for practical application. The mismatch between the computational design of nanostructures and control of bottom-up nanomanufacturing constitutes the main barrier toward fully exploiting the potential of quasi-random NMS for low-cost, high-performance devices [16,18,20].

Existing computational design methods for NMS fall into three categories, i.e., topology optimization method [5,6], descriptor-based design method [21,22], and correlation function based design method [23]. Topology optimization method pixelates the design space and demonstrates the strength in designing metamaterials and other periodic NMS [5,6]. However, as illustrated in Fig. 1, the deterministic characteristic of such design representation results in the use of expensive, often infeasible, top-down fabrication techniques.

Compared with topology optimization that involves thousands or millions of design variables, the descriptor-based method [21,22] employs an efficient parametric design approach to optimize a small set of structure descriptors, such as volume fraction, minimum distance between particles, and aspect ratio of cluster, that characterize the composition, dispersion, and geometry of NMSs. While being capable of representing NMS with regular shapes of nano- and microstructures such as the sphere-type in Fig. 2(c), the descriptor-based method has difficulty in capturing more complex morphology such as the channel-type in Fig. 2(d). Moreover, identifying the key set of descriptors usually requires complicated data analysis, such as using the machine-learning techniques [24,25].

While correlation function is a popular microstructure characterization approach [26], it is often not suitable for simulation-based microstructural design. First, optimization-based microstructure reconstruction for achieving target two-point correlation is too expensive to be included in the iterative microstructure design process. Second, model coefficients in a correlation function do not have explicit physical meaning neither possess one-to-one mapping to nanomanufacturing process conditions. Hence, optimized microstructures using the correlation function approach may not be achievable in fabrication.

We propose a novel approach for designing quasi-random NMSs, which employs a new SDF based methodology to concurrently [4] optimize the quasi-random NMS and the corresponding nanomanufacturing conditions of a bottom-up process. Previous research in image processing suggests that the Fourier spectrum, i.e., the structural Fourier transformation, is sufficient to represent any complex heterogeneous NMS and the exact material distribution can be fully reconstructed using phase recovery techniques [27]. Naturally formed quasi-random NMSs from bottom-up processes possess the isotropic nanostructures illustrated in Fig. 2 and can be sufficiently represented using a SDF. In this paper, an SDF-based approach for isotropic NMS design is developed to bridge the gap between processing–structure–performance relationship, to offer a significantly reduced dimensionality for fast design explorations and to incorporate the feasibility and stochastic nature of manufacturing processes. This paper focuses on overcoming two main technical barriers of using the SDF approach: (1) To efficiently reconstruct the real-space nanostructure in a design loop and (2) to demonstrate the feasibility of direct mapping between SDF and bottom-up nanomanufacturing conditions.

This paper is organized as follows: The SDF-based representation of quasi-random NMS is first introduced by comparing it with that of two-point correlation function (Sec. 2). The SDF-based reconstruction algorithms are then developed, demonstrating the flexibility of this method in handling different morphologies, such as the channel-type and the particle-type (Sec. 3). The SDF-based method for optimizing functional quasi-random NMS is illustrated using the optimization of quasi-random nanophotonic light-trapping structures based on structure–performance simulations (Sec. 4). Finally, the concurrent design methodology is developed by incorporating the fabrication conditions into processing–structure mapping based on the SDF representation. This new methodology is applied for optimizing a high-performance quasi-random light-trapping nanostructure fabricated using scalable wrinkle nanolithography process.

## Spectral Density Function Based Representation of Quasi-Random Nanostructured Materials

We propose to use the SDF, a one-dimensional function as the normalized radial average of the square magnitude of structural Fourier transformation, to represent isotropic quasi-random NMSs in reciprocal space. For a two-dimensional quasi-random NMS as illustrated in Fig. 3(a), the two-phase material distribution can be modeled as a function Z(r) with value 0 or 1 at each location denoted by r. The Fourier spectrum of the quasi-random NMS in Fig. 3(a) is displayed in the inset. It can be obtained by taking the Fourier transformation of Z(r) as given in the below equation

Fig. 3
Fig. 3
Close modal
$F{Z(r)}=∫ℝnZ(r)e−2πir⋅kdk=Ak⋅eiϕk$
(1)
where Ak and ϕk represent the magnitude and phase information at each location k of the Fourier spectrum. Thus, SDF can be calculated by taking the radial average of the square of the Fourier spectrum regarding k, as shown by the below equation
$f(k)=(Ak⋅eiϕk)⋅(Ak⋅e−iϕk)/C$
(2)

where k is the spatial frequency calculated as the magnitude of k as k = |k|, and C is a normalizing constant that ensures the integral of f(k) over the considered spatial frequency domain equal to one. SDF of the NMS in Fig. 3(a) is shown by the blue curve in Fig. 3(e), corresponding to the ring-shaped Fourier spectrum in the inset of Fig. 3(a).

SDF essentially describes the distribution of the Fourier components over the frequency range. The mathematical connection between the Fourier spectrum and the two-point correlation function of real-space structures has been established in literature [28]. Alternative to the two-point correlation function that describes the structural correlation in real-space, SDF characterizes structural correlation in a reciprocal space. Nevertheless, compared with two-point correlation, SDF provides a more convenient representation for designing quasi-random NMS. For example, the SDF in Fig. 3(e) can be simply formulated as the red-dashed “step” function to statistically represent the quasi-random structure in Fig. 3(a), whereas the two-point correlation in Fig. 3(i) requires a complex formulation. It is noted that the imperfectness of the step-shape of SDF in blue as compared with the red-dashed perfect one is due to the discrete pixels in the finite element meshing in numerical calculations. This imperfectness can be reduced by increasing the meshing quality.

Moreover, Figs. 3(a) and 3(b) display two quasi-random nanostructures with different feature scales. While the significant difference in the feature scale of these two structures is clearly represented by the shifting of their step-shaped SDFs over the spatial frequency as shown in Figs. 3(e) and 3(f), it is much more difficult to capture the difference between two-point correlation functions in Figs. 3(i) and 3(j). Figure 3(c) shows a more disordered quasi-random nanostructure than Fig. 3(b), which has a donut-shaped Fourier spectrum as displayed by the inset. The increased structure disorder implies the weakening of the structural correlation and less concentrated distribution of Fourier components at certain spatial frequency ranges. This difference is captured by the widening of the step-shaped SDF with lowered height in Fig. 3(g) as compared with Fig. 3(f). In contrast, two-point correlation represents the increased structure disorder by the weakened oscillation of the function as shown by comparing Fig. 3(k) with Fig. 3(j), which is less intuitive. Finally, Fig. 3(d) inset shows quasi-random nanostructures with a spread Fourier spectrum, implying the further increased structure disorder compared with Fig. 3(c). This structure is represented by the SDF as shown in Fig. 3(h). Such SDF can be formulated as a Gaussian distribution function denoted by the red-dashed curve. The different shapes of the SDFs, i.e., the Gaussian-shaped one in Fig. 3(g) and the step-shaped one in Fig. 3(h), clearly indicate the different characteristics of the structural correlation in the real-space structures in Figs. 3(c) and 3(d). However, such difference is less obvious in terms of the representation based on the two-point correlation function as shown in Figs. 3(k) and 3(l), except for the reduced oscillation in Fig. 3(l).

In conclusion, SDF provides a convenient and informative design representation of isotropic quasi-random NMS. In most cases, it enables more straightforward mathematical formulations compared with two-point correlation function. Real-space structures with various morphological characteristics can be efficiently constructed based on a given SDF for micro/nanostructure mediated design. The SDF-based reconstruction algorithms will be presented in Sec. 3. Since characteristics of the underlying structural correlation of quasi-random NMS are mostly determined by the physics of the bottom-up manufacturing process, the relation between SDF and processing conditions can be conveniently established for concurrent structure and processing design. This concurrent design methodology will be described in Sec. 4.

## Quasi-Random Structures Reconstruction Based on Spectral Density Function

Efficient reconstruction of real-space structures is critical for the automated computational design of functional quasi-random NMSs. In this section, we illustrate efficient analytical reconstruction methods based on SDF representation for quasi-random structures with different types of morphology, the channel-type and the particle-type. Though demonstrated for 2D NMSs, the approaches presented can be easily extended to 3D reconstructions.

### Channel-Type Quasi-Random Structures Reconstruction Based on SDF Using Gaussian Random Field (GRF).

The channel-type quasi-random NMSs are common nanostructures originating from bottom-up processes, such as the biological NMS in Fig. 2(b) born through the phase separation of the keratin–air mixture or the man-made NMS in Fig. 2(d) from the mechanical self-assembly of thin-film wrinkling. Gaussian random field (GRF) modeling, originally proposed for studying the spinodal-decomposition based phase separation of material mixture [29], provides a natural way to describe the NMS with such morphology.

A GRF Y(r) has two Gaussian properties: (1) the realizations of a field at randomly chosen locations in space follow a normal distribution and (2) for a fixed location r, all the possible realizations of the field at this specific point also follow a normal distribution. A standard GRF has each point marginally and all the points jointly following a standard Gaussian distribution. A standard GRF over an n-dimensional space can be fully characterized by the field–field correlation function g(r1,r2) in Eq. (3) [28,30]
$g(r1,r2)=E[Y(r1)Y(r2)]=∫0∞J(n−2)/2(kΔr)(kΔr)(n−2)/2⋅kn−1⋅f(k)dk,ΔrΔr=|r1−r2|$
(3)
$Z(r)={1Y(r)≤α0,Y(r)>α$
(4)

In Eq. (3), J are Bessel functions of the first kind, and f(k) is the SDF. The analytical relationship between the SDF f(k) and the field–field correlation g(r1,r2) that governs the Y(r) is shown in Eq. (3). Thus, the GRF Y(r) is completely described by the SDF f(k).

In numerical implementation, a realization of the GRF for a targeting SDF is constructed using the wave-form method [31] as $yr=(2/N)∑i=1N cos (kiki⋅r+ϕi)$. In this equation, N, the number of items in the truncation series, is chosen as 10,000 to ensure sufficient accuracy, ϕi is uniformly distributed over (0,2π), ki is a vector uniformly distributed on a unit sphere, and ki is a scalar distributed over (0, +∞) following the probability density function as f(kk. Once a realization of the GRF is generated, a real-space, two-phase, quasi-random structure Z(r) shown by Eq. (4) can be obtained by level-cutting the realization at α that is determined by the filler volume fraction ρ as $(1/2π)⋅∫−∞α exp (−ζ2/2)dζ=ρ$. Figure 4 illustrates the reconstruction process of a two-dimensional channel-type quasi-random NMS based on the SDF as a Delta function in Fig. 4(a).

Fig. 4
Fig. 4
Close modal

For a targeting SDF with arbitrary form, the quasi-random NMS can be efficiently constructed using this method. For further illustrations, the SDF in Fig. 5(a) follows a truncated Gaussian distribution. The constructed structure in Fig. 5(d) with the material filling ratio as 40% has a Fourier spectrum (in the inset) corresponding to the targeting SDF. Figure 5(b) shows that an SDF has two-step shape, which leads to the constructed structure in Fig. 5(e). Here, the material filling ratio is set at 50% and the structure possesses the double-band Fourier spectrum that matches with the targeting SDF. Moreover, the triple-Delta SDF in Fig. 5(c) produces the structure in Fig. 5(f) with a triple-ring Fourier spectrum. The material filling ratio is set as 60% for this case. Even though one reconstruction is illustrated for each target SDF in Fig. 5, the GRF approach also allows multiple reconstructions of statistically equivalent nanostructures if the assessment of stochastic behavior is needed. This SDF-based reconstruction method is analytical which ensures the feasibility of microstructure computational that involves iterative loops.

Fig. 5
Fig. 5
Close modal

It should be noted that in using Gaussian random field to model the isotropic quasi-random nanostructured materials, the structures are assumed to be statistically homogeneous and isotropic. The use of Gaussian random processes has three basic assumptions: (1) stationarity (independence of time), (2) isotropy, and (3) zero mean. It is also discussed in literature that there might be certain degree of spectral distortion during the level-cutting process as a nonlinear transformation of the GRF [32,33], which means the SDF of a final structure may deviate from the SDF of the original GRF. For significant spectral distortion, iterative approach can be adopted for distortion correction [34,35]. In our reconstruction shown above, limited degree of spectral distortion is observed. Since introducing such iterative correction would significantly impact the efficiency of an iterative microstructure optimization process, the distortion correction is not applied in this work. Moreover, it should be noted that there are various random field methods for structure reconstruction, such as these discussed in Refs. [32,34], and [36]. Interested readers may refer to literature for more details.

### Particle-Type Quasi-Random Structures Reconstruction Based on SDF Using Random Packing.

Particle-type NMS is another important class of quasi-random structures that have the potential of being fabricated by bottom-up processes, such as the NMS in Fig. 2(c) made by the self-assembly of polymer and cuttlefish ink nanoparticles. For particle-type NMS, the spatial correlation depicted in SDF is determined by particle sizes and distances between particles. In this work, we develop an algorithm based on random disk packing to achieve nonoverlapping particle-type NMS for a given SDF profile. Existing works have investigated the possibility of packing equal-sized disk shape particles to achieve certain types of SDF by adjusting the center distances and number of disks. Most of these structures are loosely packed and possess a narrow ring shape SDF [37]. In this work, we establish a semi-analytical relationship between distributions of disk numbers and disk radiuses with any arbitrary target SDF profiles; the method is also applicable for close packing structures.

The close disk packing algorithm we propose follows a simple strategy as shown in Fig. 6(c): if two neighboring disks are found to be too overlapped, then both will be pushed apart; vice versa if the distance between two neighboring disks is too large (not desired for dense packing), then both will be moved toward each other. The appropriate distances to move are calculated using the formula in the below equation

Fig. 6
Fig. 6
Close modal
$Δ1=r2r1+r2(r1+r2−d), Δ2=r1r1+r2(r1+r2−d)$
(5)

where Δ1 and Δ2 are the moving distances for disks 1 and 2, respectively, and the spatial adjustment is along the line connecting the centers of the two disks. A positive value means the two disks will move closer and negative value means they will move away from each other. d is the current center distance between two disks, and r1 and r2 are the radius of the two disks, respectively. To realize a close disk packing, we start from a randomly packed structure that allows overlapping, then adjust the position of each disk in the structure until there is no overlapping in the structure and all the disks are touching its neighbors within a small allowance ε (d = r1 + r2 ± ε). By choosing different disk sizes, we are also changing the spatial distances between neighboring disks. For a given number and the sizes of the disks, this strategy is found effective in generating close disk packing structures with volume fraction up to 90%. It is noteworthy that we only require disks center to be within the target region instead of the whole disks so that we can reduce large voids near the boundaries of the structure. This disk-packing algorithm can be extended to generate microstructures with particles of other shapes. The centers of particles are determined first by the disk-packing algorithm, and then the particles of other shape are assigned to those centers. In this way, the spatial distances between particles in the new microstructure are controlled by the disk size chosen in the packing algorithm. For SDF-based reconstruction, the challenge is how to bridge the gap between the structure and its SDF.

The case of equal-size close disk packing is first studied and then extended to the packing of multiple-size disks for arbitrary SDF profiles. SDF describes a structure in the frequency domain and measures the underlying periodicity of the structure, a feature closely related to the number of disks packed. The typical SDF of an equal-size random close disk packing structure in Fig. 6(a) resembles dense hexagonal packings and has apparent periodicity. Thus, its SDF in Fig. 6(b) has its components concentrated on a single ring (corresponding to a single frequency kpeak as marked in the figure). Through our empirical studies, in which we generated multiple equal-size disk-packing structures using different disk radiuses (from L/40 to L/20, where L is the side length of the structure), we discovered that the radius of the ring in Fig. 6(b) can be estimated based on the number of disks packed in the structure: kpeak = 1.1√N, where N is the total number of disks, and this relationship is obtained by simple regression analysis for periodic square packing and hexagonal packing, similar relationship exists despite the structure size. It is noteworthy that the relationship for equal-size random close packing is very similar to that of hexagonal packing, which can be justified by the fact that hexagonal packing is the densest packing for equal-size disks and if we pack the disks closely, they tend to form hexagonal lattices. In addition, for close packing of equal-size disks, the number of disks N can be estimated based on disk radius r by considering its similarity to hexagonal packing N = VF*L2/πr2, where ρ ≈ 0.9069. By substituting N into the expression of kpeak, we can establish the simplified relationship between kpeak and disk radius r
$r≈L/1.69kpeak$
(6)

where L is the side length of the square-shaped structure. This relationship indicates that by adjusting the size of disks, we can change the frequency of SDF.

More complex SDF profiles can be achieved by packing disks with different sizes. For simplicity, a uniformly distributed SDF (see Fig. 3(e)) is used as an example to illustrate the strategy. Assuming the target SDF is uniformly distributed between a frequency interval [k1, k2], the appropriate range of disk sizes can be determined by Eq. (6). The problem becomes how to choose the right number of disks for each size so that we can achieve a uniform SDF. This problem is addressed through optimization in this work by minimizing the fluctuation of SDF within the target range [k1, k2] and adjusting the number of disks for each size. Although optimization can provide satisfactory results based on our tests, it is computationally prohibitive, especially after we add the outer loop for searching the optimal SDF in material design.

To overcome the computational difficulty, we again employ approximated relations derived from empirical studies. We create multiple unequal-size packing structures by varying the number of each size and find that the intensity of SDF f(k) at a certain frequency ki is proportional to the total area of the corresponding disk of radius ri, where ki and ri satisfy the relationship in Eq. (6)
$f(ki)∝Niri2$
(7)
where Ni is the number of disks of radius ri. In addition, when the total areas of disks for each size ri are equal, the corresponding f(ki) are equal. As a result, to achieve a uniform SDF, the following condition holds:
$N1r12=N2r22=⋯=Nnrn2$
(8)
from which the number of disks for each size ri can be derived
$Ni=Anπri2=0.9069L2nπri2, i=1,2,...,n$
(9)

where n represents the number of different disk sizes, A is the total covered area by disks, and L is the side length of the square structure. The constant 0.9069 is the volume fraction of equal-size hexagonal disk packing and here we use it as an approximation of the volume fraction in our packed structure and it can be updated for different target volume fractions. Here, we found that the empirical relation works well for forming a uniform SDF. We can adjust the frequency range k of SDF f(k) by choosing appropriate disk sizes using Eq. (6) and then adjust the value of SDF at each frequency k by changing the numbers of disks of each size using Eq. (7). Therefore, SDF with an arbitrary shape can be approximated in this way. Three examples are shown in Fig. 7 to verify the effectiveness and versatility of our approach: the SDF in Fig. 7(d) consists of two narrow step functions and the corresponding NMS in Fig. 7(e) possesses two apparent feature scales. Figures 7(b) and 7(c) are NMSs with the same underlying SDF in Fig. 7(a) that follows a broad Gaussian distribution where the particles have different sizes. This observation is consistent with the relationship between particle size and frequency in Eq. (6). It is noted that although the particle geometries are totally different in Figs. 7(d) and 7(e) (circle and triangle), the underlying spatial correlation is essentially the same, which is determined by the number of particles and spatial distances. In conclusion, by integrating the disk-packing algorithm and the empirical relationships established in this section, multiple statistically equivalent particle-type NMS can be efficiently constructed toward any given arbitrary SDF with few limitations on particle geometries.

Fig. 7
Fig. 7
Close modal

## Spectral Density Function Based Computational Design of Quasi-Random Nanostructured Materials

Conventional real-space design approach directly employs the pixelated material distribution within the design space as the design representation where each pixel is a design variable. Such strategy leads to thousands or millions of design variables and thus numerical issues associated with high design dimensionality. Most importantly, the deterministic nature of such real-space structure representation is incapable of capturing the nondeterministic characteristics of quasi-random NMSs made by bottom-up processes.

In this paper, we propose a new SDF-based concurrent design methodology for optimizing quasi-random NMS and the corresponding manufacturing process as shown in Fig. 8. The key innovation of the concurrent design strategy is to use SDF f(k) as the design representation to enable the processing–structure mapping. Instead of using the pixelated real-space material distribution as design variables as in topology optimization, the SDF of the quasi-random structure plus a few additional structure descriptors, such as the material filling ratio, are regarded as the design variables. We have shown in Sec. 2 that SDF provides a convenient and informative design representation in reciprocal space for quasi-random structures with complicated real-space geometries. With our approach, given the desired performance, the specific types of the SDF that match with the nanomanufacturing process are first derived. Based on the SDF and a few additional structure descriptors, the quasi-random NMSs in real space are efficiently constructed using the methods discussed in Sec. 3 for simulations to assess material properties and device performance. For given design objectives, this computational design loop iterates to identify the optimal design of quasi-random nanophotonic structures in reciprocal space. By utilizing this SDF representation, only 4–5 design variables are needed for a typical problem to preserve nondeterministic characteristics of quasi-random NMS.

Fig. 8
Fig. 8
Close modal

Since the physics of a bottom-up process determines the structural correlation in fabricated quasi-random NMSs, a mapping between a bottom-up process and a specific form of SDF can be derived either through physics-based modeling or empirical studies, which enables the concurrent structure and processing design of functional quasi-random NMS. Specifically, once the SDF formulation is optimized for targeting functional performance using the computational design methodology in the right side of Fig. 8, the corresponding process conditions are identified via the mapping between processing conditions and the SDF design representation shown at the left side of Fig. 8. This concurrent design methodology ensures the feasibility of the optimized structure for the chosen bottom-up nanomanufacturing process.

In Sec. 4.1, the computational design of light-trapping NMS is presented first as an example to illustrate the effectiveness of using SDF as design representation. Next, in Sec. 4.2, the optimization of quasi-random light-trapping nanostructure fabricated using nanowrinkle-based process is presented as an example to show how manufacturing processing conditions of nanowrinkling are mapped to SDF for achieving concurrent design of structure and processing conditions.

### Spectral Density Function Based Computational Design of Quasi-Random Light-Trapping Nanostructure for Thin-Film Solar Cells.

A light-trapping structure in a thin-film solar cell is employed as a representative example to demonstrate the SDF-based computational design methodology for quasi-random NMSs. Thin-film solar cells with significantly reduced thickness of absorbing layer possess unique advantages, such as flexibility, semi-transparency, and low usage of expensive material in the absorbing layer, as compared with bulk cells [38]. However, the thinner absorbing layer in thin-film solar cells leads to less interaction between the incoming light and the active material, hence lower energy absorption efficiency. Light trapping was therefore developed to extend the path-length for light interacting with the active material, so highly efficient thin-film solar cell can be created. Figure 9(a) illustrates a simplified thin-film solar cell model, where the top layer is the light-trapping nanostructure, gray color denotes amorphous silicon (a-Si) as the active material for light absorption, and the bottom red layer represents a silver layer for preventing the light escaping on the back side. The total thickness of the absorbing layer made of a-Si is t.

Fig. 9
Fig. 9
Close modal
To couple the incoming light into the quasi-guided modes supported by the absorbing layer requires the appropriate structural correlation of the quasi-random light-trapping nanostructures. Meanwhile, the material filling ratio ρ of the quasi-random nanostructure, which determines effective refractive index of the top-layer, also significantly influences the light absorption efficiency. Since the quasi-random light-trapping structure as shown in Fig. 9(a) is made of a-Si, the optimal light-trapping effect also depends on the depth of the quasi-random nanostructure t1. The large depth of the light-trapping nanostructure usually produces strong scattering effect for light coupling, which however reduces the total volume of active material. In this case, the SDF f(k) with the material filling ratio ρ and the depth t1 are regarded as the design representation of the quasi-random light-trapping nanostructure to be optimized using the methodology outlined in Fig. 8. The design objective is shown in the below equation
$maxZ(f(k),ρ,t):A(Z(f(k),ρ,t1))$
(10)

where Z represents the quasi-random nanostructures, and A is the absorption coefficient as the ratio between the energy absorbed in the cell and the total energy of the incoming light. Based on quasi-random light-trapping nanostructure constructed from the design representation [f(k), ρ, t1] with other material properties and structure parameters, the whole device is modeled for the performance simulation. Here, we use the rigorous coupled wave analysis (RCWA) method [39,40] for performance simulation. RCWA is a Fourier space based algorithm that provides the exact solution of Maxwell's equations for the electromagnetic diffraction by optical grating and multilayer stacks structures. Other methods such as finite-difference time-domain (FDTD) or finite element analysis (FEA) can be applied for the performance simulation, as well. To account for the stochasticity embedded in the reconstruction process, the average absorption coefficient of three reconstructed structures is calculated and treated as the objective function for the SDF-based design optimization. Based on the simulation results, the design variables are updated using optimization search algorithms. While different algorithms can be adopted for design updating, here we use the genetic algorithm (GA). Mimicking natural evolution with the underlying idea of survival-of-the-fittest, GA is a stochastic, global search approach involving the iterative operation of selection, recombination, and mutation on a population of designs [41,42]. The stochasticity of GA enables the convergence toward optimums despite the strong nonlinearity in a design problem [5].

For the purpose of demonstration, here we assume that the SDF follows a step function as shown in the inset of Fig. 9(b), which is governed by two variables, i.e., ka and kb. Thus, together with the material filling ratio and the depth of the quasi-random light-trapping nanostructures, this design problem involves four (4) design variables, i.e., ka, kb, ρ, and t1. Optimization for a single incident wavelength w = 700 nm, which corresponds to the weak absorption region of a-Si, is first performed to verify the efficacy of the methodology. The total thickness of the a-Si layer is set as 600 nm, and the length p of the RCWA simulation window is set as 2000 nm. The optimization history is shown in Fig. 9(b), in which each dot represents the corresponding absorption coefficient for a given set of design variables (ka, kb, ρ, t1). Starting from the randomly generated initial designs with the absorption coefficient around 0.12, the optimization converges to the solutions with the absorption coefficient larger than 0.79. Statistical representation using SDF enables infinite numbers of optimized quasi-random structures to be generated in real space. Six designs were constructed with an identical set of optimized variables, denoted by the red dot in Fig. 9(b), with the channel-type structures in Figs. 9(c)9(e) generated using Gaussian random field modeling and the sphere-type structures in Figs. 9(f)9(h) generated using random disk-packing, respectively. The optimized design denoted by the red dot is $ka*$ = 0.0029 nm−1, $kb*$ = 0.0030 nm−1, ρ*= 62%, and $t1*$ = 75 nm. Despite the different real-space morphologies, all the designs possess similar ring-shaped Fourier spectrum (shown in the insets of Figs. 9(c)9(h)) and achieved the equal optimal performance. For further validation, shown in Fig. 9(i), 50 real-space patterns were generated from the optimized design and evaluated to achieve an average absorption of 0.80 with less than 4% variation; the variation is due to the discrete meshing in numerical calculation, thereby achieving a 4.7-fold enhancement of the absorption compared to the uniform (unpatterned) structure.

A broadband optimization of the quasi-random light-trapping nanostructures over the visible solar spectrum is then conducted using our SDF-based computational design methodology. Broadband optimization over the solar spectrum from 400 nm to 800 nm (w = 400–800 nm) is considered here. The structural parameters in this case are set to be t = 650 nm and p = 2000 nm. In this optimization, 81 wavelengths are considered over the whole wavelength spectrum. The objective is to maximize the absorption enhancement factor which is the ratio between the predicted absorption and the single-path absorption averaged over the considered wavelengths [43], as shown in Eq. (10). In this equation, Z denotes the nanostructure to be optimized described by design variables [f (k), ρ, t1], wi denotes the ith considered wavelength, A is the predicted light absorption of design Z, and AS denotes the single-path light absorption of a uniform cell as a normalizing factor independent of the design
$max:Z∑wiwnA(Z,wi)AS(wi)/n$
(11)

The broadband optimized design is $ka*$ = 0.0012 nm−1, $kb*$ = 0.0036 nm−1, ρ*= 51%, and $t1*$ = 150 nm, which leads to the quasi-random structure with a donut-shaped Fourier spectrum as shown by the insets of Figs. 10(a) and 10(b). This Fourier spectrum overlaps with the area defined by two dashed circles that denote the range of k-vectors for coupling the incident light to the quasi-guided modes in the silicon film [19,44]. Two distinct light-trapping designs in real space are realized from the identical optimized SDF, which is a channel-type structure in Fig. 10(a) and a particle-type structure in Fig. 10(b), respectively.

Fig. 10
Fig. 10
Close modal

The optimized structures are compared with the two reference structures that are commonly adopted in literature [45] for improving light-trapping performance. The two reference structures include an unpatterned cell with a 70 nm silicon dioxide antireflection coating (ARC) and an unpatterned cell without the ARC, both of which have the same thickness of absorbing material (a-Si) as the optimized ones. The absorption spectra of the two optimized quasi-random structures and two reference cells are calculated using RCWA and plotted in Fig. 10(c). Serving simultaneously as an efficient quasi-guided mode coupler and an ARC, the two optimized quasi-random light-trapping nanostructures with drastically different real-space morphology enhanced the absorption by more than threefold in the weak absorption region (600–800 nm) compared with the two references, achieving an average absorption coefficient of 0.74 over the broad spectrum from 400 nm to 800 nm. As shown in this testbed, designing the structure in reciprocal space using SDF-based representation instead of tailoring the structure in real-space provides the freedom to down-select the optimized real-space structure in order to better accommodate manufacturing constraints. For example, the channel-type structure can be fabricated using the nanowrinkling process as illustrated in Fig. 2(d), and the sphere-type structure can be fabricated using nanoparticle self-assembly as illustrated in Fig. 2(c). Furthermore, the fabrication conditions can be determined through the inherent connection between SDF and the physics of a bottom-up process. This will be discussed using nanowrinkling-based fabrication process as an example in Sec. 4.2.

### Concurrent Design of Quasi-Random Light-Trapping Nanostructure for Scalable Nanomanufacturing Process Based on Wrinkle Lithography.

We have demonstrated the computational design of functional quasi-random NMS using SDF-based methodology, where we assume that the SDF to be optimized follows a step function. While bottom-up process based nanomanufacturing provides a scalable fabrication method for realizing quasi-random NMS designs, the fabrication process has not been fully considered in the design shown in Figs. 9 and 10. In this section, an example of incorporating the conditions of the fabrication process into the design stage is provided for the design of quasi-random NMS fabricated using thin-film nanowrinkling.

Wrinkling of the thin layer has emerged as a simple method for the scalable fabrication of microscale and nanoscale surface structures. In this bottom-up nanomanufacturing method, a stiff skin layer on a softer, prestrained substrate is buckled during the relaxing of the substrate, forming the wrinkle nanostructure as shown in Fig. 11(a). The primary wavelength of the resulted wrinkles λw is linearly proportional to the thickness of the skin layer. In this work, the skin layer is chosen as the prestrained, thermoplastic polystyrene (PS). The thickness of the PS layer is precisely controlled by changing the reactive ion etching (RIE) time of CHF3 plasma gas ($TCHF3$) in processing for continuously tuning λw. The primary winkle wavelength of the structure shown in Fig. 11(a) is λw = 180 nm.

Fig. 11
Fig. 11
Close modal

To utilize wrinkling process for fabricating functional quasi-random NMS, the nanowrinkle structure in Fig. 11(a) needs to be transferred to other materials. In our research, we transferred the nanowrinkle onto the a-Si via a nanolithography process for the light-trapping purposes. The transferred nanowrinkle pattern forms 2D quasi-random nanostructures. In this process, the PS wrinkles are treated with SF6 plasma and then transferred into a PDMS mask. The a-Si thin-film is coated with a layer of Al2O3, upon which a photoresist is spin cast. Solvent-assisted nanoembossing (SANE) [46] is completed with the wrinkled PDMS mask and photoresist protects wafer, resulting in the original wrinkle pattern being transferred into photoresist. The application of timed direction O2 plasma in RIE reduces the thickness of the photoresist. The time of the application of O2 plasma ($TO2$) in RIE determines the material filling ratio ρ of the resulting structure.

After this wet etching process, the photoresist is removed and the 3D wrinkle structure has now been transferred to an Al2O3 mask. The Al2O3 can then act as a deep reactive ion etching (DRIE) mask for etching into the a-Si without affecting the λw of the wrinkle feature, generating the final quasi-random nanostructure for light-trapping. The time of DRIE determines (TDRIE) the depth of the quasi-random light-trapping structure, i.e., t1 as shown in Fig. 9(a).

While it is challenging to model the resulting quasi-random nanostructure from nanowrinkle patterning by using real-space representation, the SDF-based method provides a convenient representation of the structure. As shown in Fig. 11(b), the blue curve is the SDF of the scanning electron microscope (SEM) image of patterned structure from the nanowrinkle shown in Fig. 11(a). The peak position of the measure SDF, i.e., km, is dependent on the primary wrinkle wavelength λw as km = 1/λw. By empirically analyzing the five fabricated samples with different primary wrinkle wavelengths, we derive that the SDF of the transferred wrinkle pattern follows a truncated Gaussian distribution function. Figures 11(b)11(d) show the analyzing results for three different samples of wrinkle pattern (λw = 180 nm, 450 nm, and 2000 nm). In each of the figures, the blue curve is the SDFs of the SEMs and the red-dashed curve is the fitted SDF that has the truncated shape from a Gaussian distribution function governed by only one variable as μ = km. The standard deviation of the Gaussian distribution depends on km as σ = 0.958 μ + 0.00017. The r-square for the fitting is 0.99. Here, we normalize the truncated Gaussian distribution to the integral of one over the range 0–0.02 nm−1 for Fig. 11(b), 0–0.01 nm−1 for Fig. 11(c), and 0–0.005 nm−1 for Fig. 11(d), respectively. It is noted that the tails of the fitted SDFs do not match well with those from SEMs in Figs. 8(b)8(d). This is caused by a number of small features which are either due to imaging noises or small manufacturing defects. These small features correspond to the Fourier components at higher frequency, leading to the fat tails of SDFs. The prominent features of the nanowrinkle patterned structure are characterized by the “hill” region of the SDFs. In this case, despite the slight mismatch at the tail region, our fitted SDFs capture the main features of the nanowrinkle patterned structure.

Utilizing the SDF-based representation, the wrinkle patterned quasi-random nanostructure contains only three variables, i.e., km = 1/λw, ρ, and t1. Based on our data from the processing testing, these three design variables can be independently controlled, following relations as $TCHF3$ = 0.2087·λW − 37.5758, $TO2$ − 36.86·ρ + 71.63, and TDRIE = 0.325·t1 − 2.7451. This processing-design mapping enables the concurrent design outlined in Fig. 8.

Here, the Gaussian random field modeling is used to construct the real-space structure for the performance simulation. The total thickness of the a-Si material is set as 700 nm in this case. The optimized design for the broadband light-trapping over 800–1200 nm wavelength range is $km*$ = 0.0018 nm−1, ρ*= 52%, and $t1*$ = 210 nm. The optimized structure significantly enhances the absorption compared with the unpatterned uniform structure, achieving 150% absorption enhancement over the unpatterned thin-film in the weak-absorbing region of the material (800–1200 nm).

The optimized structure is then fabricated using the wrinkle nanolithography, according to the processing-design mapping. It is found that the fabricated optimal wrinkle pattern achieves 130% absorption enhancement over the unpatterned thin-film in the weak-absorbing region of the material (900–1200 nm). The SDF of the SEM of fabricated sample matches well with the optimized SDF design. This testbed validates the efficacy of using the novel concurrent design methodology for realizing high-performance quasi-random NMS while designing and ensuring the feasibility of the bottom-up scalable nanomanufacturing process.

## Conclusion

In this paper, a novel nondeterministic representation is proposed for designing quasi-random nanostructures with inherent robustness. A SDF based concurrent design method is developed for designing cost-effective and scalable quasi-random NMSs and the associated bottom-up nanomanufacturing processes. As a representation of NMSs in the reciprocal frequency space, SDF captures the spatial correlation at different length scales. Compared with the widely used two-point correlation function, SDF is shown to be more-effective in differentiating quasi-random NMS. Moreover, SDF generally takes simple forms that can be related to physical processing conditions, which bridges the gap between processing–structure relation that facilitates the concurrent structure and processing design. To enable automated computational microstructure design, reconstruction techniques are developed in this work for creating statistically equivalent quasi-random NMSs based on a given SDF. Efficient reconstructions are achieved for both the channel-type NMS systems and particle-type NMS systems using the Gaussian random field (GRF) modeling and the random disk-packing based approach, respectively. The results show that the mapping between SDF and nanostructure is not unique—although the real-space morphologies are completely different, the NMSs of these two systems may share common SDFs as long as their essential spatial correlations are similar. The SDF-based design strategy provides the freedom to down-select optimized design from different real-space structures to accommodate the constraints of manufacturing platforms. Although the NMSs presented in this paper are isotropic, our approach can be extended for anisotropic structures by modeling the GRF using anisotropic correlations for quasi-random NMs and using nonuniform orientation distributions for particle-type NMSs.

To verify the effectiveness of using SDF as a design representation for quasi-random NMS, a light-trapping structure in a thin-film solar cell is employed as an example. The optimized structure with the ring-shaped SDF achieved 4.7-fold enhancement of single length absorption and for broadband light-trapping, the SDF of the optimal structure is uniformly distributed, with two (2) times overall improved performance. Designing the structure by SDF rather than real-space morphologies increased the freedom of down-selecting different NMS systems (e.g., channel-type and particle-type) to accommodate manufacturing constraints. We further perform a concurrent design of quasi-random light-trapping structure fabricated using the wrinkling nanolithography-based scalable nanomanufacturing technique. The realistic fabrication condition of the wrinkle nanolithography process is integrated into the design stage by deriving the SDF of the wrinkle patterns as a truncated Gaussian distribution function. This SDF along with other structure representation variable is optimized to identify the optimal wrinkle pattern for light-trapping purposes with the fabrication feasibility for wrinkle lithography process.

We believe that this SDF-based method can well solve the structure design problems of the functional materials whose performance largely depends on the first- and second-order spatial correlations of the structures. These problems are abundant in various areas, such as the design of noniridescent structural coloration coating [16], the material with excellent transport property [47], or the nanodielectric material with high permittivity and low dielectric loss [48]. This research contributes to the creation of a new and accelerated materials system design paradigm with the emphasis on achieving the compatibility between nanostructure design and the design of scalable nanomanufacturing processes. The research will be further enhanced by testing the approach on different material systems with different desired functional performance and nanofabrication processes.

## Acknowledgment

The grant supports from the U.S. National Science Foundation (NSF) EEC-1530734 and CMMI-1462633 are greatly appreciated. The authors would like to thank Dr. Clifford Engel and Dr. Alexander Hryn at Northwestern for their assistance on fabricating the nanowrinkle structures the optical performance measurement and also Dr. Zhen Jiang currently at the Ford Motor Company on the discussion of using Gaussian random field modeling for structure reconstruction. S. Yu and W. K. Lee also thank the International Institute for Nanotechnology for the Ryan Fellowship Award.

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