Stealthy Hyperuniform Surface Structures for Efficiency Enhancement of Organic Solar Cells

Low absorption in the thin active layer of conventional organic solar cells limits their power conversion efficiency. Structured surface layers are a common approach to diffracting incoming light, thus elongating its path through the active layer, thereby increasing the probability of absorption and hence the power conversion efficiency. While standard periodic structures diffract light into discrete angles, making them optimal only for specific wavelengths, random structures induce broadband, but nontailorable diffraction. Thus, instead, a stealthy hyperuniform structure, designed to exhibit beneficial diffraction properties is implemented: it directs the light into a predefined range of higher angles, prevents diffraction into small angles, and is thus ideal for a strong active path length enhancement. After numerical optimization of the feature height and diameter, the stealthy hyperuniform structure is fabricated in silicon by electron beam lithography and subsequently transferred into a transparent polymer via replica molding. Experimental diffraction images reveal a circular symmetric spectrum, inducing diffraction independent of the azimuthal angle and polarization of the incident light. The application of the stealthy hyperuniform structure on a poly[(2,6‐(4,8‐bis(5‐(2‐ethylhexyl)thiophen‐2‐yl)‐benzo[1,2‐b:4,5‐b′]dithiophene))‐alt‐(5,5‐(1′,3′‐di‐2‐thienyl‐5′,7′‐bis(2‐ethylhexyl)benzo[1′,2′‐c:4′,5′‐c′]dithiophene‐4,8‐dione)]:3,9‐bis(2‐methylene‐(3‐(1,1‐dicyanomethylene)‐indanone))‐5,5,11,11‐tetrakis(4‐hexylphenyl)‐dithieno[2,3‐d:2′,3′‐d’]‐s‐indaceno[1,2‐b:5,6‐b′]dithiophene organic solar cell leads to a sharp increase in current density and power conversion efficiency.

inherently cannot be adapted to the solar cell layout or absorption spectrum. [11]herefore, more intricate, aperiodic structures have recently been investigated as potential alternatives.Similar to random structures, they are suitable for broadband absorption enhancement, but can also be tailored to the specific solar cell configuration and materials.So far, such surface structures have been developed mainly for gallium arsenide (GaAs) and silicon solar cells.For instance, quasirandom nanostructures inside GaAs cells have been numerically engineered to couple the incident light into waveguide modes to increase the absorption. [12,13]urthermore, also the top part of silicon substrates has been designed quasirandomly or with a biomimetic micro-and nanostructure resembling the surface of a black butterfly to redirect light into waveguide modes, resulting in an experimentally demonstrated absorption enhancement. [14,15]ere, a stealthy hyperuniform structure is chosen for its characteristic spatial frequency spectrum, which induces diffraction into a defined angular range and thus offers high potential for an optimized broadband absorption enhancement in solar cells.The structure belongs to the larger group of hyperuniform systems first introduced in 2003 by Torquato and Stillinger in the context of many-particle systems. [16][19][20] These structures have already been implemented to create unique electromagnetic scattering at interfaces that could be used to construct colored surfaces in architecture and design or for spectral molecular fingerprint detection. [18]In addition, stealthy hyperuniform photonic materials have been constructed that exhibit complete bandgaps blocking all polarizations and directions. [17,21]Finally, the top part of a silicon substrate has already been structured according to a stealthy hyperuniform pattern, which led to improved absorption. [22]Accordingly, for a complete silicon solar cell an increase in photovoltaic efficiency was predicted.However, this efficiency gain due to the absorption enhancement is reduced by surface-induced charge carrier recombination at the structures.
In order to circumvent this challenge, in this contribution a stealthy hyperuniform surface phase structure is designed.As it is simply added on top of the cell, it has no negative impact on charge carrier extraction and allows facile comparison of cell efficiency with and without surface structure.The stealthy hyperuniform pattern is generated by sequentially altering a random jammed pattern to finally exhibit a Fourier transform without small frequencies, thus preventing the corresponding phase structure from diffracting light into small angles.The generated phase structure is then numerically optimized for a poly[ ]dithiophene (ITIC) organic solar cell using the simulation method outlined in our previous publication. [23]The stealthy hyperuniform phase structure consists of a microstructure with features on the order of 500 nm that diffract light at larger angles.However, reducing the size of the structure to the nanometer range would result in an antireflection surface. [24]Therefore, one could, in general, extend the stealthy hyperuniform structure to a hybrid structure with locally different scaling that takes advantage of both: antireflection properties and path length elongation.The microphase structure is fabricated by first creating a silicon master using electron beam lithography and reactive ion etching, and subsequently transferring the structure into a transparent polymer by replica molding.Finally, the produced stealthy hyperuniform phase structure is evaluated based on its diffractive characteristics and its influence on the PCE of a PBDB-T:ITIC organic solar cell.

Stealthy Hyperuniform Pattern Generation
According to the Fraunhofer equation, the diffraction pattern in the far field of a phase structure is defined by its Fourier transform (FT). [25]Thus, the distance from the origin of the FT defines the angle at which the light is diffracted.A high intensity near the origin means that the light is diffracted at small angles, resulting in only a small increase in path length through the active layer and, accordingly, only a small absorption enhancement.In contrast, a high intensity at greater distances from the origin signifies diffraction at larger angles and a corresponding greater path length elongation.However, too large diffraction angles can induce total reflection at one of the solar cell interfaces, which is detrimental to the absorption enhancement if it occurs before the light enters the active layer.Circular symmetry in the Fourier transform signifies that incident light is diffracted independent of its azimuthal angle, which is advantageous for continuous absorption enhancement of sunlight whose azimuthal angle changes throughout the day.
Stealthy hyperuniform structures are defined by their structure factor, which is proportional to the absolute square of their Fourier transform. [26]They belong to the broader group of hyperuniform systems, all of which have a structure factor SðkÞ that satisfies the condition: [27] lim Thus, the structure factor tends to zero as the wave vector k approaches zero.This means that the mean-square structural fluctuations increase less rapidly than R 2 , where R is the radius of the observation window. [17]ll crystals and quasicrystals belong to this group of hyperuniform systems, but some disordered systems also fulfill the above condition. [28]They do not exhibit Bragg peaks in their diffraction spectrum, but also no large-scale density fluctuations, indicating a hidden symmetry.
A special type of hyperuniform system is the stealthy hyperuniform one.Their structure factor is circular symmetric and equal to zero for wave vectors k smaller than the exclusion wave vector k ex around the origin: [17,21] Accordingly, stealthy hyperuniform surface structures diffract light exclusively at larger angles, which can greatly enhance the active path length of a solar cell. [28]Furthermore, having a circular symmetric FT, they diffract light independent of its azimuthal angle.Thus, they meet all the basic requirements for an optimal surface structure.
In order to ultimately obtain a stealthy hyperuniform pattern, a random jammed point pattern is generated first (see Figure 1a).It is created by placing points one after another at a random position in a square.If there is an overlap with an existing point, the new point is discarded.To create a highly jammed pattern, each time before a new potential point is created, all existing points are randomly shifted by at most one pixel in any direction.In case of an overlap with another point, they are moved back to their original position.Points touching the boundaries of the square are bounced back.
The size of the individual points is adjusted so that the fill fraction of the pattern with its 49% is very close to 50%.This fill fraction is targeted because it proved to lead to the highest diffraction efficiency into the first diffraction order for periodic diffraction gratings. [29]he absolute square of the Fourier transform of the final random jammed pattern is plotted logarithmically in Figure 1c.It exhibits a symmetric circle of high intensity around the origin and thus already exhibits the isotropy targeted for the FT of the stealthy hyperuniform pattern.
The stealthy hyperuniform pattern is created by displacing a random point in the random jammed pattern by at most one pixel and calculating the new absolute square of the FT.If it has lower intensity within a circle with the previously defined exclusion wave vector k ex around the origin, the modified pattern is used.Otherwise, the original pattern is kept.The exclusion wave vector is chosen to match the circle already having a slightly lower intensity in the random jammed pattern.By optimizing the real sizes of the individual points in a later step, the higher intensity region then corresponds exactly to the wave vectors needed for optimal diffraction in our organic solar cell.The process of displacing individual points is repeated 500 000 times and is limited only by the computation time.Thus, the stealthy hyperuniform pattern shown in Figure 1b with its FT shown in Figure 1d is obtained.As it is constructed by simply shifting the points of the random jammed pattern, the stealthy hyperuniform pattern also has a fill fraction of 49%.
A possibility to test the hyperuniformity of a pattern is to calculate how the number variance σ 2 N ðRÞ ≡ NðRÞ 2 h iÀ NðRÞ h i 2 of particles within a spherical observation window scales with its radius R. [30] If the number variance grows slower than the window volume, i.e., slower than R 2 , the pattern is hyperuniform.The number variance of the stealthy hyperuniform pattern constructed here grows slightly slower than R 1.8 , thus proving the hyperuniformity of the pattern.

Numerical Optimization
The stealthy hyperuniform surface structure is optimized for the solar cell depicted in Figure 2. It is composed of a silver (Ag) anode, a molybdenum oxide (MoO 3 ) hole-transport layer, a bulk-hetero-junction active layer, a zinc oxide (ZnO) electrontransport layer, an indium tin oxide (ITO) cathode, and a glass substrate.
The active layer materials consists of the high-performance donor polymer PBDB-T and the small molecule acceptor ITIC.Solar cells based on these materials have already achieved efficiencies of over 12%, making them a good starting point for further optimization through surface structures. [31]The phase structure itself consists of the transparent polymer polydimethylsiloxane (PDMS) and is attached to the glass substrate.As both the surface layer and the solar cell have an approximate thickness of 1 mm, the active layer lies in the far field of the phase structure.Thus, as mentioned above, the diffraction pattern in the active layer is defined by the mean square of the Fourier transform of the field directly behind the phase structure.
Figure 2 shows a simplified illustration of the diffraction of a plane wave with wavelength λ incident on the stealthy hyperuniform phase structure on top of the solar cell.The diffraction angle induced by the phase structure changes as the light passes through the different solar cell layers with their different refractive indices.Finally, in the active layer, the oblique angle of incidence of the light leads to a path length elongation P compared to the straight light path (dashed line).
The optimization of the stealthy hyperuniform structure is done numerically based on the calculations outlined in the publication by Merkel et al. [23] The individual steps are described in detail in the Supporting Information.In short, first, the active path length elongation induced by the stealthy hyperuniform surface structure is calculated for each wavelength.Then, the corresponding current density increase is determined, taking into account the absorption and the solar spectrum as well as multiple reflections in the active layer, adjacent layers, and at the back electrode.The structure factor SðkÞ of the generated stealthy hyperuniform phase structure leads to a current density increase of 4.2%.Finally, the pillar diameter and height are optimized sequentially to achieve the highest current density increase.Accordingly, for a pillar diameter of 620 nm and a height of 590 nm, a current density increase of 4.6% is calculated.

Fabrication Process
The fabrication process of the optimized stealthy hyperuniform surface structure consists of two main steps.First, the generation of a master structure in a silicon substrate by electron beam lithography and reactive ion etching, and second, the replica molding of the structure into the transparent polymer PDMS.
Hence, a silicon substrate is first coated with an adhesion promoter and subsequently with a high-resolution negative-tone e-beam resist.It is then exposed using an electron beam lithography system to generate the stealthy hyperuniform structure in the resist.As shown in the supporting information, the optimal pillar diameter of the structure is 590 nm.However, a deviation to smaller diameters caused by inaccuracies in e-beam writing would greatly diminish the current density increase induced in the solar cell.Thus, a slightly larger diameter of 620 nm is used instead because it lies on the plateau of high current density increase shown in Figure S3a, Supporting Information.In this case, small deviations will not have a large effect on the final performance of the surface structure.Based on this selected pillar diameter, a single pattern shown in Figure 1b has a size of 41.9 μm Â 41.9 μm.Therefore, it is repeated 53 Â 48 times to generate a total pattern of 2.2 mm Â 2.0 mm, which can eventually cover the entire solar cell.
After exposure, the resist is developed to remove the unexposed areas.In order to transfer the resulting structure in the resist to the silicon substrate, the samples are then treated with reactive ion etching.The etch parameters applied should result in an approximate etch depth of 700 nm.The details of the fabrication are described in Section 8 and an image of the process flow is included in the supporting information.The final structures in the silicon substrate are depicted in Figure 3.
Overall, the structures closely resemble the intended stealthy hyperuniform pattern.There are only some imperfections in form of small indentations.In addition, the individual pillars are slightly too large with a diameter of about 638 nm.Therefore, the fill fraction of the structure is not 50% as planned, but 58.3%.
In the second part of the fabrication, the structure of the silicon master is transferred to the transparent polymer PDMS by replica molding.For this purpose, the silicon master is made hydrophobic by silanization with trimethylchlorosilane (TMCS) to reduce the affinity of the PDMS and facilitate its detachment at the end. [32]Then a mixture of the PDMS base and its curing agent is poured over the silicon master and cured in an oven.The amount is adjusted so that the final PDMS layer is 1 mm thick.After cooling, the PDMS film can be peeled off the silicon substrate and be used as a solar cell surface structure.An atomic force microscope image of part of this structure is depicted in Figure S4, Supporting Information.It corresponds to the negative of the silicon master.However, the depth of the structure is not uniform but varies in the range from 100 to 550 nm.Nevertheless, the current density in the solar cell is expected to be enhanced for all of these depths (see Figure S3b, Supporting Information).

Diffraction Patterns
To verify the predicted diffraction properties of the stealthy hyperuniform surface structure, its diffraction pattern is recorded using the setup depicted in Figure 4a. [33,34]Here, 405 nm laser light is focused onto the PDMS film using a lens with a focal length of 210 nm.To illuminate only the structured part of the film, a mask is placed directly in front of the sample, leaving only a 2.0 mm Â 2.2 mm large gap.The polarization of the light is linearly adjusted with a half-wave (λ/2) plate.Additionally, to obtain circularly polarized light, a quarter-wave (λ/4) plate can be placed before the lens.Behind the sample, the light is collected by a high NA = 1.0 microscope objective (water immersion, 60Â).The light is then collimated and directed onto a CCD camera via a 4f system with two lenses with focal lengths of 75 and 100 mm.Thus, the Fourier transform of the surface phase structure is imaged on the camera.Between the two lenses, a pinhole acts as a field stop, such that only light from the structured region of the sample is collected.
In Figure 4b the measured diffraction pattern is depicted.Next to it, in Figure 4c, the simulated diffraction pattern of the optimized structure is shown.Unlike in Figure 1d, the intensity is plotted linearly rather than logarithmically.In addition, the simulated results are plotted in the same wave vector range as the measurement.The relationship between Fourier space and pixel position was scaled by placing a transmission grating with 300 lines mm À1 in place of the stealthy hyperuniform sample and measuring the distance between the diffraction maxima.In both simulation and experiment, the diffraction pattern is circularly symmetric, with a ring of higher intensity around a (slightly) darker center.The inner radius of this ring is 7.5 μm À1 , which corresponds to the characteristic average next-neighbor center-tocenter distance between the pillars of the stealthy hyperuniform structure. [18]or finite diffractive structures, a peak at zero wave number of the diffraction pattern remains, signifying the nondiffracted light.In the measurement, it is discernible as a bright spot in the center, while in the simulation the central pixel is set to zero to improve the visibility of the rest of the pattern.In the remaining part of the inner circle, the measured diffraction pattern is also brighter than the simulated one.This is partly due to the slightly higher fill fraction of the fabricated structure of 58.3%.For comparison, in Figure 4d the simulated diffraction pattern of the optimized structure assuming a fill fraction of 61% is plotted.The pattern is slightly brighter in the center and has a thinner ring around it than the one in Figure 4c.An additional reason for the brighter center is the manufacturing imperfections of the final PDMS sample (see Figure S4, Supporting Information).These explain the higher intensity around the center because their formation is random and therefore amplifies structural long-range fluctuations. [18]he measured diffraction pattern exhibits a pixelated character, which is due to the repetition of the single stealthy hyperuniform pattern to generate a larger structure.This repetition is equivalent to a convolution of the individual structure with a 2D comb structure.Thus, the diffraction pattern corresponds to a multiplication of the Fourier transforms of the single pattern and the comb.
In order to experimentally demonstrate polarizationinsensitive diffraction, the diffraction patterns are measured for horizontal, vertical, and circular polarization of the incident light.Figure 4e-g shows that the diffraction pattern does not change with varying polarization.Such polarization insensitive diffraction has also been observed for other structures with circular symmetric diffraction patterns before, e.g., the Vogel spiral. [33]It is of advantage for solar cell surface structures because they, thus, diffract all parts of the unpolarized sunlight equally.

Effect on Solar Cell Parameters
After experimentally studying the diffraction properties of the stealthy hyperuniform surface structure, its effect on a solar cell is investigated.For this purpose, the current density-voltage ( J-V ) curve of a PBDB-T:ITIC cell, having the layout shown in Figure 2, is measured multiple times with and without surface structure.Its PCE, short-circuit current density ( J sc ), and opencircuit voltage (V oc ) are rather low a priori, in contrast to the optimal PBDB-T:ITIC cell assumed in the simulation.Thus, there is a much higher potential for improvement in the experiment, making it rather a proof of principle for the effect of the stealthy hyperuniform surface structure than a mean to quantify its potential.
The J-V curve of the cell is first measured 30 times without the surface structure, then 30 times with the structure, and then again 30 times without the structure.This repetition allows to obtain statistically valid results despite possible fluctuations in the values.The time between each measurement is 10 s and the application and removal of the surface structure between the three measurement sections takes about 5 min.
The evolution of the PCE, short-circuit current density ( J sc ), and open-circuit voltage (V oc ) obtained from the J-V curves is plotted in Figure 5.For both PCE and J sc , the values in each section first increase slightly for about 70 s and then decrease slightly.However, the change due to the application and removal of the surface structure is much more pronounced.
This change is quantified by calculating the mean value with and without surface structure, respectively.In the case of the PCE, this results in an increase of (61 AE 8)% caused by the surface structure.For the J sc , an increase of (41 AE 5)% and, for the V oc , an increase of (7 AE 2)% are calculated.The measurement uncertainty of each of these values is determined from the standard deviations of the values with and without surface structure.

Conclusion
In summary, we demonstrated the generation, numerical optimization, fabrication, and application of a stealthy hyperuniform surface structure for the efficiency enhancement of an organic solar cell.The structure was generated by shifting the points of a random jammed point pattern until its circular symmetric Fourier transform yields a low-intensity circle around the origin.The diameter and height of the pillars were then numerically optimized to achieve the strongest active path elongation and thus the highest current density increase in a PBDB-T:ITIC solar cell. [23]or fabrication of the surface structure, the optimized structure was first generated in silicon using electron beam lithography and reactive ion etching.It was then transferred into the transparent polymer PDMS via replica molding to obtain the final phase structure.The diffraction pattern of this stealthy hyperuniform surface structure consists of a brighter ring around a darker center.Thus, incident light is diffracted in a fixed angular range corresponding to the higher intensity ring.As the pattern is circular symmetric, diffraction is independent of both the azimuthal angle and the polarization, which simplifies universally beneficial application under sunlight.
Finally, its usefulness as a surface structure was demonstrated by applying it to a PBDB-T:ITIC solar cell.The stealthy hyperuniform surface structure induced a sharp increase in PCE, short-circuit current density, and open-circuit voltage, making it a valuable tool in organic solar cell optimization.By tailoring the angular range into which most light is diffracted, stealthy hyperuniform structures also have the potential to be optimally adjusted to the absorption spectra of various other types of solar cells.Large-area structures can be realized using commercially available deep UV lithography systems for the fabrication of the master structure, followed by replica molding into PDMS.

Experimental Section
Fabrication of Stealthy Hyperuniform Master Structure: Prior to electron beam lithography, the 525 μm thick silicon substrate was cleaned by sonication in acetone for 5 min, rinsed with isopropanol, and treated in an O 2 plasma cleaner for 5 min.For better adhesion of the e-beam resist, the adhesion promoter AR 300-80 new (Allresist) was spin-coated onto the silicon substrate at 67 s À1 for 60 s and subsequently annealed at 180 °C for 2 min.Then, the high-resolution negative-tone e-beam resist AR-N 7520 (Allresist) was spin-coated on top at a rate of 50 s À1 for 90 s and annealed at 85 °C for 90 s.The resist was exposed using an EBPG 5150 electron beam lithography system (Raith GmbH) with an area dose of 1600 μCcm À2 , a beam current of 50 nA, and a write time of 7 h, and developed by immersion in Microposit MF-319 developer (Kayaku Advanced Materials) for 60 s.Then, the sample was etched with reactive ion etching in a PlasmaPro 100 ICP device (Oxford Instruments) to transfer the structure to the silicon substrate (RF power: 50 W, ICP power: 300 W, SF 6 flow: 5 sccm, CHF 3 flow: 60 sccm, pressure: 15 mTorr, temperature: 20 °C, duration: 7 min).The remaining resist was removed by O 2 plasma cleaning, and the resulting structures were examined by scanning electron microscopy using a CrossBeam 340 system (Zeiss).
Replica Molding into PDMS: The silicon substrate was made hydrophobic by silanization with TMCS to reduce the affinity of PDMS and facilitate its detachment at the end.Meanwhile, a Sylgard 184 PDMS base (Dow Corning) was mixed with its curing agent at a ratio of 10:1 vol% and subsequently kept in vacuum for 20 min to degas any bubbles formed during mixing.Then, the PDMS was poured over the silicon master, baked at 80 °C for 2 h, and cooled down before the final PDMS surface structure was peeled off.
Solar Cell Fabrication: ITO-coated glass substrates (Präzisions Glas & Optik GmbH, 100 nm ITO thickness) were cleaned by ultrasonication in deionized water, acetone, and isopropanol for 10 min each.Afterward, they were treated for 10 min in an O 2 plasma cleaner to improve the wettability of the subsequent ZnO layer.The liquid ZnO precursor solution was prepared by mixing 0.5 g zinc acetate dihydrate and 0.14 g monoethanolamine in 5 mL methoxyethanol and stirring overnight.Then, 100 μL of the ZnO precursor solution was spin-coated at 67 s À1 for 40 s on the ITO layer and annealed at 200 °C for 30 min.For the preparation of the active layer, PBDB-T (Ossila Ltd.) and ITIC (Luminescence Technology Corp.) were mixed with chlorobenzene at a concentration of 10 mg mL À1 each.
The solution was then stirred at 60 °C for at least 12 h.About 1 h before spin-coating, 0.5 vol% of 1,8-diiodooctane was added to the solution.When the ZnO layer had cooled, 150 μL of the active material solution was spin-coated on top at 42 s À1 for 60 s to achieve a layer thickness of about 100 nm, and was subsequently annealed at 70 °C for 10 min.Finally, 10 nm of MoO 3 and 100 nm of Ag were deposited by thermal evaporation.

Figure 2 .
Figure 2. Scheme of light diffraction by stealthy hyperuniform phase structure on an organic solar cell with reflective back electrode.The incident light of wavelength λ is diffracted by the phase structure at an oblique angle, which changes as it passes through the different solar cell layers with their different refractive indices.The light's path is shown here exemplarily for a diffraction angle of 30°behind the phase structure and a wavelength λ of 400 nm.For this wavelength, the individual layers have the following refractive indices: n PDMS ¼ 1.41, n Glass ¼ 1.52, n ITO ¼ 1.95, n ZnO ¼ 1.69, n BHJ ¼ 1.60, n MoO 3 ¼ 1.96, and n Ag ¼ 0.19.The oblique incidence of the light into the active layer results in a path length elongation P compared to straight incidence (dashed line).Multiple reflection and refraction are indicated by the thinner arrows.

Figure 1 .
Figure 1.Surface structures and the absolute square of their Fourier transforms, which is proportional to their respective structure factor.a) Random jammed pattern.b) Stealthy hyperuniform pattern.c,d) Corresponding Fourier transforms plotted logarithmically.

Figure 3 .
Figure 3. Scanning electron microscopy of the stealthy hyperuniform structure in silicon.

Figure 4 .
Figure 4. a) Experimental setup for measuring far-field diffraction patterns of the stealthy hyperuniform structure illuminated with 405 nm laser light.λ/2: half-wave plate, L: lens, S: stealthy hyperuniform structure with mask, MO: infinity-corrected microscope objective, P: pinhole, and Cam: camera.b) Measured diffraction pattern.c) Simulated diffraction pattern of optimized structure.d) Simulation assuming a fill fraction of 61% of the optimized structure.e) Measured diffraction pattern for horizontal polarization.f ) Measured diffraction pattern for vertical polarization.g) Measured diffraction pattern for circular polarization.

Figure 5 .
Figure 5. Solar cell parameters with and without the surface structure.The three marked sections are the same for each parameter: a) PCE, b) J sc , and c) V oc .