In this paper, the role of pseudo-disordered moth-eye structures on the optical features for application to thin-film solar cells is investigated to realize the superior light management for the full-spectrum solar energy utilization, compared with some ordered structures. Without loss of generality, the c-Si thin film solar cell is taken as the example. The results demonstrate that the fluctuations introduced into the geometry parameters of moth-eye elements can lead to the remarkable absorption enhancement in the wavelength region of 0.3-1.1 μm and high transmission in the wavelength range of 1.1-2.5 μm. Two mechanisms including the increasing spectral density of modes and the intensive forescattering intensity are identified to be responsible for the absorption enhancement. In addition, the optical characteristics of the moth-eye surface with both disordered height and disordered diameter are insensitive to the incident angle.
© 2017 Optical Society of America
CorrectionsXiaojun Liu, Yun Da, and Yimin Xuan, "Full-spectrum light management by pseudo-disordered moth-eye structures for thin film solar cells: erratum," Opt. Express 25, A870-A870 (2017)
As fossil energy sources are depleting and the global warming and environment pollution become more and more serious, researchers pay more and more attentions to the clean and renewable solar energy. Among all types of approaches to utilize solar energy, converting solar energy into electric energy is one of most scientific and sensible ways.
Photovoltaic cells are the main optoelectronic devices which can acquire electric energy from solar energy immediately by photovoltaic (PV) effect. For various kinds of PV cells, c-Si thin film solar cells with less material consumption are one of most promising cost-effective PV cells [1,2]. For thin film solar cells, an inevitable drawback is the poor optical absorption due to the thin active layer [2–13]. In recent years, various ordered micro/nano-patterns, such as gratings [4,5], photonic crystals [6,7], plasmonics [3,8] and bio-inspired structure [8–13] assembled on the front, rear or both sides of thin film solar cells, have been proposed to solve it. The absorption enhancement in the infrared band is still insufficient, although solar cells with these optimized structures show a higher absorption, compared with planar solar cells.
One modified approach is to adopt disordered structures [14–18] or pseudo-disordered structures [19–24] instead of ordered structures. It has been demonstrated that these complex structures are more efficient for trapping incident photons. In these investigations, it is widely believed that the absorption enhancement should be attributed to the intensified guided resonance. However, these studies hardly give a deeper analysis of the guided resonance in theory. Moreover, it is difficult to simulate disordered structures due to high degree of inorganization .
On the other hand, these investigations on Si solar cells only focus on the wavelength below 1.1 μm which is restricted by the band-gap of silicon. As is well-known, the energy of the solar spectrum is substantially concentrated in the wavelength range of 0.3-2.5 μm and the photons beyond 1.12 eV account for about only 80% of the whole solar energy . It is a pity that most investigations neglect the solar energy in the wavelength range of 1.1-2.5 μm, which holds about 20% of the whole solar energy.
Photovoltaic-thermoelectric (PV-TE) hybrid systems are regarded as one of most promising strategies to achieve the utilization of the full-spectrum solar energy [26–31]. It has been demonstrated in theory that PV-TE hybrid systems can increase the efficiency by at least 8% . In order to realize the high-efficiency PV-TE hybrid system which is achieved by simply stacking the TE module on the back side of the PV module, a key requirement for Si solar cells is to achieve appropriate light management. The core concept of the light management is as follows: silicon based surfaces should enhance the absorption in 0.3-1.1 μm and keep the high transmission in 1.1-2.5 μm . Among various structures, bio-inspired moth-eye structures are suitable for the light management because of the outstanding antireflection feature [8,30–32]. The main mechanism to reduce reflection is the near linear refractive index gradient between air and active layer, according to the effective medium theory [8–13,31,33]. So the high aspect ratio of moth-eye elements usually means great antireflection features [8,9,31]. However, although the great antireflection effect of moth-eye structures general means high absorption, the effect of resonances and scattering on the absorption of thin film solar cells is also unnegligible .
In this article, we propose a pseudo-disordered moth-eye nanostructure to achieve the light management based on Si thin film solar cells of PV-TE hybrid systems. Firstly, via the finite-difference time-domain (FDTD) method, we preliminarily optimize the ordered moth-eye structures as references and analyze the reason of the absorption enhancement based on the electric field distribution of ordered moth-eye surfaces corresponding to different wavelengths. Then the fluctuations are introduced into moth-eye structures. In this section, we investigate the effect of various pseudo-disordered moth-eye structures, such as height-disordered structures, diameter-disordered structures and position-disordered structures, on the light management. In addition, in order to reveal the physical mechanisms of the absorption enhancement, we discuss the characteristics of forescattering and guided resonances for pseudo-disordered structures by analyzing the forescattering efficiency and bandstructure, respectively. Moreover, the influence of incident angle on the optical properties of silicon solar cells is investigated in the last section.
2. Structure model and calculation method
2.1 Simulation model
The sketch map of PV-TE hybrid systems is displayed in Fig. 1(a). One can see that the TE module is simply stacked in the rear of the PV module. It is assumed that the all photons penetrating through solar cells are absorbed by the TE module. Thus, we focus on the optical characteristics of the PV module in this paper, without considering the influence of the TE module. The previous work from our group has demonstrated that the application of a back Si/SiO2/air interface can effectively enhance the transmission in 1.1-2.5 μm and keep the absorption characteristics of the PV module in 0.3-2.5 μm almost changeless at the same time . In this work, in order to expressly exhibit the optical characteristics of moth-eye structures, the simple c-Si thin film solar cells without back interfaces are employed for the PV module and consist of Si moth-eye structured surfaces and Si membranes. The geometry parameters of a unit cell of moth-eye structures are given in Fig. 1(b). The parameters consist of the period , the height h1, the diameter D and the membrane thickness h2. In this paper, the thickness of the silicon membrane is fixed at h2 = 1.5 μm. In general case, the position of the centroid of a moth-eye element is placed in the center of corresponding unit cell. In order to simulate the pseudo-disordered structure in FDTD, the supercell is proposed to convert the disordered pattern into the ordered one. As shown in Fig. 1(c), a supercell consisting of 33 unit cells is regarded as a simulation period for both disordered structures and ordered structures  and the corresponding geometric parameters are added to every unit cell in a supercell. It is worth noting that the fluctuation range of the geometric parameters of diameter, height and position meets the uniform distribution. The profile section of the moth-eye element is parabola and the mathematic description of the shape can be expressed as 
2.2 Simulation method
The FDTD method is applied to calculating the spectral characteristics of our proposed structures. The simulation is achieved by the software of Lumerical FDTD solutions . In the process of the simulation, the material properties are obtained from Palik . For normal incidence, the periodic boundary conditions are applied in the x- and y- directions while the perfect matched layer (PML) boundary conditions are applied in the z- direction. It is appropriate to only simulate the pseudo-disordered moth-eye structures under the s-polarization normal incident light [18,24] because of the central symmetry of the moth-eye element. For oblique incident, the periodic boundary conditions are replaced by Bloch boundary conditions in the x- and y- directions because of the phase difference of the incidence light in a simulation period. When the reflection, transmission, absorption and electric field distribution are calculated, the plane wave source is utilized. It is worthy noticing that the plane wave source is replaced by the dipole source to calculate the bandstructure. In addition, in view of the reliability of the spectral characteristics of the pseudo-disordered moth-eye structure, we adopt 8 times simulations and 12 times simulations for one disordered parameter and two disordered parameters respectively and regard the average value as the final result.
In order to quantitatively evaluate the optical properties of the moth-eye structure, the integral reflection, transmission and absorption are adopted and described as 
When the light impinges on the object, a portion of light is scattered in the forward hemisphere, which refers to the forescattering phenomenon. The forescattering efficiency defined as the ratio of forescattering cross section to geometry cross section is commonly utilized to evaluate the scattering performance of moth-eye surfaces. The total-field scattered-field (TFSF) source substitutes the plane wave source to compute the forescattering efficiency. The forescattering power and forescattering efficiency is expressed as [8,37]
3. Results and discussion
3.1 Optimization of ordered moth-eye structures
The period is one of most crucial parameters of the moth-eye element for the light management of silicon thin film solar cells. In order to elucidate the superiority of the moth-eye structured surface, the flat Si which means the planar c-Si thin film solar cell without surface structures is utilized as a reference. As shown in Fig. 2(a), the influence of the period on the optical property of solar cells is preliminarily investigated by scanning this parameter from 0.3 μm to 0.5 μm, in order to explore the influence of the difference between the period and the diameter. One can see that the absorption of the optimized period of 0.4 μm exhibits a surprising rise of 39.5%, compared with the flat Si. It should be noticed that the volumes of silicon material in both considered systems are the same and the thickness of the flat Si is 1.677 μm. As the period increases, it is clear that the graded refractive index (GRI) at the interface between moth-eye structures and membranes is more and more discontinuous. This results in the fast increment of the reflection in 0.3-1.1 μm and the sustained decline of the transmission in 1.1-2.5 μm. In addition, the increasing period also enhances the guided resonance, which is helpful for the absorption of sunlight. To further optimize the diameter and height of moth-eye structures with a period of 0.4 μm, as shown in Figs. 2(b) and 2(c), we scan the diameter from 0.25 μm to 0.35 μm and the height from 0.3 μm to 0.9 μm, respectively. One can see that a suitable size for the diameter and height is 0.3 μm and 0.8 μm, respectively. Figure 2(d) shows a comparison of the absorption spectrum between the optimized moth-eye ordered structure and the flat Si. The smooth line appears in the wavelength range of both 0.3-0.7 μm and 1.15-2.5 μm, while lots of sharp absorption peaks occur in the remaining region. This reveals that there are three different main mechanisms of the enhancement corresponding to three wavelength regions, respectively. Detailed discussions are written as below.
To understand the optical characteristics of silicon moth-eye structures, we respectively calculate the electric field distribution of silicon cells with the period of 0.5 μm at the wavelengths of 0.57 μm, 1.03 μm and 1.75 μm, as depicted in Fig. 3. From Fig. 3(a), one can see that the incident light can hardly penetrate the active layer of the c-Si due to the high extinction coefficient at the short wavelength band. This means that the absorption at this wavelength band is determined by the ability of antireflection and forescattering of structured surfaces, whereas the increasing period weakens this ability. As shown in Fig. 3(b), the light penetrates through the membranes and is coupled with a Bloch mode [38–40], which results in an unusual high absorption peak similar to that in Fig. 2(d). As shown in Fig. 3(c), the electric field distribution exhibits a noticeable interference fringes in the region of the Si membrane and the corresponding reflection trough appears. The main reason is that when the wavelength is much larger than the diameter, the moth-eye surface acts like a flat Si layer and the optical spectra are determined by the Fabry-Perot interference. The reflection peaks can be determined by the interference condition :
3.2 Effect of the disordered geometry parameters of pseudo-disordered moth-eye structures
3.2.1 Effect of the disorder in diameter
The effect of the disorder in the diameter of moth-eye elements on the optical properties of silicon cells is firstly investigated to find the maximum absorption in 0.3-1.1 μm and high transmission in 1.1-2.5 μm. The height and the period are respectively fixed at 0.8 μm and 0.4 μm, based on the preliminary optimized result in the last section. The disordered diameter is realized by assigning randomly values to every unit in a supercell. In order to quantitatively describe the pseudo-disordered moth-eye structure, the disordered diameter can be defined as
The ordered structure with the reference diameter is optimized during the fluctuation range of diameter. In addition, the fluctuation of diameter results in little change of the volume. Hence, it is reasonable to select the diameter-optimized ordered structure as a reference. Figure 4 shows a comparison of optical properties between the ordered surface and the pseudo-disordered structure. The fluctuation range of 0 μm, 0.05 μm and 0.1 μm is selected for the comparison. It is found in Fig. 4(a) that as the increases, the reflection in the wavelength range of 0.3-0.6 μm is suppressed, which is in consistent with the GRI theory. More specifically, the break of the gradient of GRI at the interface between moth-eye surfaces and membranes is weak when the fluctuation range increases, so that the reflection is reduced. Furthermore, the smaller the parameter is, the more distinctly the Fabry-Perot interference phenomenon appears. As seen in Fig. 4(b), it can be concluded that the integral absorption of the pseudo-disordered structure in 0.8-1.1 μm has been greatly enhanced compared to the ordered structure. This results from that the disordered array highlights the coupling between incident light and additional modes, which intensifies the guided resonance [34,38]. It is worth noting that the more expected peaks for diameter-disordered structures are hard to be recognized. This is because the density of modes is so large that some guided resonant peaks with greater amplitude suppress the adjacent absorption peaks with smaller amplitudes. Figure 4(c) provides the average integral reflection, transmission and absorption in the wavelength range of both 0.3-1.1 μm and 1.1-2.5 μm. As the fluctuation range increases from 0 μm to 0.1 μm, the absorption in 0.3-1.1 μm rises by 5.1%. Therefore, the diameter-disordered structures are beneficial for the light management.
3.2.2 Effect of the disorder in height
In this subsection, we optimize the disorder degree of the height. Similar to the disordered diameter, the height in each unit cell of a supercell is allocated to a random value while the diameter and period are both equal to 0.3 μm. The disordered height can be quantitatively expressed as
Similar to the diameter-disordered structure, the height-optimized ordered structure is regarded as a reference to illustrate the advantage of the height-disordered structure. Figure 5 collects the average optical properties at normal incident for the structure surface with three different parameters. One can observe that the highest absorption of 0.804 in 0.3-1.1 μm is achieved when the equals 0.3 μm. This indicates that on the one hand, the higher disorder degree excites more additional modes, especially at large wavelengths. On the other hand, it doesn’t mean stronger forescattering at short wavelength bands. Therefore, the optimized is 0.3 μm according to the requirement of the light management.
Through a comparative investigation of the optimized and shown in Fig. 6, we tentatively discuss the mechanism of the absorption enhancement. Figure 6(a) offers the forescattering efficiency spectra of the two structures without considering the membranes. It is worth noting that the forescattering intensity of the ordered one at the wavelengths of 0.68 μm and 0.92 μm is close to zero. This means that the scattering light transmitted in the forward hemisphere at corresponding wavelengths is weakened and the light path is reduced, resulting in the inhibited absorption as the black dot line in Fig. 6(b) shows. When the fluctuation is introduced into the height, the moth-eye structure acts as an irregular cavity and supports more complex multi-reflection behavior. As a result, the forescattering intensity at corresponding wavelengths is sharply enhanced and the light path in the Si active layer increases, which supports the rise of the absorption shown in Fig. 6(b). In addition, the disordered height can also contribute to coupling light with more additional modes, compared to the ordered one. Thus, the wider and more peaks appear in the red line of Fig. 6(b). From Fig. 6(c), it can be seen that the absorption in 0.3-1.1 μm increases by 9.85%, while the transmission in 1.1-2.5 μm reduces by only 5.5%. Thereby, the height-disordered structures are pretty suitable for the light management.
3.2.3 Effect of the disorder in both height and diameter
It has been concluded that the pseudo-disordered structures are able to realize superior light trapping due to the break of symmetry. As depicted in Fig. 7, the average optical properties of pseudo-disordered structures with both disordered height and disordered diameter (case 3) are compared with that of the aforementioned disordered height (case 1) and disordered diameter (case 2) in Fig. 7. Obvious, similar to case 2, the transmission of case 3 in 1.1-2.5 μm is much lower than that of case 1, while the absorption of case 3 in 0.3-1.1 μm is almost equal to that of case 2. The main reason is that although the introduction of two disordered parameters can increase the couple between incident light and modes, the enhancement degree caused by the guided resonance is lower than the decline of the absorption resulting from the worse GRI. This limits the further improvement of the light management.
3.3 Effect of the disordered position of pseudo-disordered moth-eye structures
Here the effect of the position with small fluctuations on the light management is studied in Fig. 8 by two comparison groups. To generate position-disordered moth-eye structures, the moth-eye elements in every unit cell of a supercell are moved from their original position with a small fluctuation, ensuring each moth-eye element limited within in the primary unit cell at the same time. The moth-eye element is shifted by a random length and the length meets the uniform distribution.
From Fig. 8(a), it is clear that the absorption in 0.3-1.1 μm of the position-disordered structure is only a little larger than that of the position-ordered structure. The main reason is that the fluctuation introduced in the position of the moth-eye structure is still small. Another reason is that although the introduction of the disordered position can enhance the guided resonances near the bandgap, it can also weaken the resonances in the short wavelength range. Similar situation also appears in Fig. 8(b). Obviously the results indicate the small fluctuation introduced into the position is an insignificant help to the improvement the light management.
3.4 Analysis of modes in the reciprocal space
In the previous sections, we have given the primary explanations on the optical phenomena of moth-eye structures. Nevertheless, the mechanism of absorption enhancement, especially caused by the guided resonance is not clear enough. In this section, a deeper interpretation is given via the comparison of the bandstructure between ordered moth-eye structures and pseudo-disordered structures.
The spectral density of the modes for the ordered moth-eye structure with μm is ultrahigh in the Brillouin zone. However, we are mainly interested in the point, since the plane wave with normal incidence can only excite the modes at the point according to the diffraction mechanism . The detailed bandstructure at the point is plotted in Fig. 9(a). In Fig. 9(a), the modes of ordered structures at the point are located in the wavelength range of 0.6μm −1.05μm. It can be seen that the incident light excites some modes and then corresponding absorption peaks appear at corresponding wavelengths, which markedly enhances the absorption of Si solar cells. More specifically, only part of modes which are symmetric with respect to the in-plane crystallographic directions are excited, whereas lots of antisymmetric modes are not coupled to normal incident light [39,42].
Based on the last analysis, the bandstructure of the height-disordered structure is discussed. Due to the supercell consisting of unit cells and the disordered fluctuations breaking the symmetry of a supercell, the spectral density of modes of the height-disordered structure is much larger than that of the ordered one, especially near bandgap. As shown in Fig. 9(b), we only give the bandstructure of the best group for the resolution of modes at the point. The modes are located in the wavelength range of 0.73μm −1.1μm. From the comparison between Figs. 9(a) and 9(b), an obvious absorption enhancement of the height-disordered structure is achieved in 0.6-0.73 μm where there is little mode. With respect to the forescattering efficiency in 0.6-0.8 μm from Fig. 6(a), it could be concluded that the increasing forescattering resulting in the increment of light path in the Si active layer plays an important role in the absorption enhancement. It is clear that the modes’ density of height-disordered structures near bandgap is larger. And this enhances the guided resonance and increases the absorption.
Figures 10(a) and 10(b) exhibit a comparison of the bandstructure between the diameter-disordered structure and the corresponding ordered structure. Similar to the height-disordered structure, the diameter-disordered structure also highlights the spectral density of modes at the wavelength near the bandgap, which is beneficial for the absorption enhancement of thin film solar cells. Obviously, more guided resonances are excited when the density of modes is high. Another notable phenomenon is that the number of absorption peaks in 0.7μm –1.1μm is not as much as expected, although the spectral density of modes for diameter-disordered structures is much higher than that of the ordered one. This phenomenon results from the fact that the absorption at some resonant spots is large enough to suppress the resonant peaks excited by nearby modes. Indeed, the disordered structure could excite more and stronger guided resonant peaks with the help of the high spectral density of modes. Therefore, both height-disordered moth-eye structures and diameter-disordered structures are consistent with the requirement of the light management that the solar cells of PV-TE hybrid systems should achieve excellent absorption in 0.3μm −1.1μm and high transmission in 1.1μm −2.5μm at the same time.
Figures 10(a) and 10(c) exhibit a comparison of the bandstructure between position-disordered structures and corresponding position-ordered structures with , and . One can see that the mode density of position-disordered structures in 0.75-1.05 μm increases and the corresponding absorption is enhanced as well. But the decline of the density in 0.6-0.75μm weakens the absorption at the same time. In addition, one can also see that the intensity of the additional resonances resulting from the small fluctuation in position is quite low. This results the poor absorption enhancement in 0.3-1.1 μm (as shown in Fig. 8).
3.5 Angular response of the moth-eye structure with disordered height and disordered diameter
In order to investigate the optical characteristics of the moth-eye structure with both disordered height and disordered diameter under oblique incidence, we calculate the average of absorption and transmission for s-polarized light and p-polarized light at three selected wavelengths of 0.6 μm, 1.2 μm and 1.8 μm, respectively.
Figure 11(a) illustrates the angular response of the absorption of the ordered surface and the surface with both disordered height and disordered diameter. It can be observed that the absorption of disordered structures at 0.6 μm reduces from 0.967 to 0.924 as the incident angle rises from 0 o to 60 o, while the absorption of the ordered one reduces from 0.992 to 0.94. This means that the antireflection feature of the disordered structures in short wavelength band is insensitive to the incident angle, being similar to the ordered structures. As depicted in Fig. 11(b), the transmission within 0° to 60° surpasses 0.5 at the wavelength of 1.8μm thanks to the effect of the forescattering. By the same token, the drop within 0° to 60° is slight although the transmission at 1.2 μm declines with the increment of the incident angle. Therefore, when the incident angle is smaller than 60°, it could be estimated that the high absorption in 0.3-1.1 μm and high transmission in 0.3-2.5 μm could be maintained for the moth-eye structure with both disordered diameter and disordered height.
In conclusion, the pseudo-disordered Si-based moth-eye structures have been studied for the utilization of full-spectrum solar energy. The corresponding light management approach requires that the Si thin film solar cells should make the absorption in 0.3-1.1 μm as high as possible, on the basis of the high transmission in 1.1μm-2.5μm. Firstly, we have demonstrated that the absorption characteristics of the moth-eye structure with the geometry parameters including disordered height, disordered diameter and both disordered height and disordered diameter are better than that of the corresponding ordered structures. The enhanced guided resonance especially excited near the bandgap is the main mechanism of the absorption enhancement and the forescattering intensity has an obvious influence on the enhancement as well. In addition, we find that the introduction of both disordered height and disordered diameter couldn’t achieve a better light management than the introduction of either disordered height or disordered diameter. Then according to the investigation of the influence of position-disordered moth-eye structures on the light management, it is found that the position-disordered moth-eye structure whose moth-eye elements are still limited in corresponding unit cells could hardly contribute to the improvement of the light management because the disordered degree is low and the weak absorption enhancement in the wavelength range of near the bandgap is neutralized by the decline of the absorption in the middle considered wavelength range. It can be also learned that the pseudo-disordered moth-eye structure with and exhibits great angle-insensitive characteristics within the incident angle smaller than 60°. Finally, such structures can be fabricated by means of the widely used nanoimprint lithography technique  and this work has provided instructive guides for improving the light management by the pseudo-disordered moth-eye structure.
National Natural Science Foundation of China (NSFC) (No. 51590901, 51336003).
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