## Abstract

We carry out a systematic numerical study of the effects of aperiodicity on silicon nanorod anti-reflection structures. We use the scattering matrix method to calculate the average reflection loss over the solar spectrum for periodic and aperiodic arrangements of nanorods. We find that aperiodicity can either improve or deteriorate the anti-reflection performance, depending on the nanorod diameter. We use a guided random-walk algorithm to design optimal aperiodic structures that exhibit lower reflection loss than both optimal periodic and random aperiodic structures.

© 2011 OSA

Minimizing the reflectance from a semiconductor surface is important for achieving high-efficiency solar cells. For silicon solar cells in particular, micron-scale pyramidal surface texturing [1] and silicon nitride anti-reflection coatings have been widely used to reduce the surface reflectance. Over the last decade, with the development of nanofabrication techniques, various novel high-performance anti-reflection schemes have been proposed and demonstrated [2] that take advantage of subwavelength surface texturing [3–12]. One approach to subwavelength surface texture is to use periodic arrays of nanostructures with various shapes, fabricated using lithography techniques. In practice, however, some degree of structural randomness is always present, due to fabrication imperfections. Moreover, while lithography-free fabrication techniques [9] can offer inexpensive, scalable methods for nanopatterning, they tend to produce structures with large structural randomness. Therefore, understanding the effect of aperiodicity on antireflection behavior has important practical implications. One work [9] has suggested that height variations in nanotip structures can improve antireflection performance. However, the effect of positional randomness on the performance of subwavelength antireflective structures has not been modeled systematically.

In this work, we use large-scale electromagnetic simulations to investigate the effects of aperiodicity on broadband reflection. Aperiodic nanorod structures can exhibit significantly better antireflection performance than periodic ones. We perform calculations of the solar-averaged reflection loss for hundreds of random configurations to characterize the statistical effects of aperiodicity. Our results show that the effects of aperiodicity vary with nanorod diameter. For smaller diameters, randomness is generally beneficial, while for larger nanorod diameters, it degrades the anti-reflection performance. We further use optimal design techniques [13] to identify optimal aperiodic structures that outperform their periodic counterparts over a range of nanorod sizes.

Figure 1 illustrates three different types of structures that we simulate. Vertical silicon nanorods are arranged periodically (Fig. 1(a)), almost periodically (Fig. 1(b)), or randomly (Fig. 1(c)) on a semi-infinite silicon substrate. In the periodic structure, the nanorods are arranged in a square lattice. In the almost periodic structure, the nanorods have a small amount of positional disorder. In the random structure, the nanorods are randomly positioned. Each of these structures will have different reflective properties. The height of the nanorod is fixed to 120 nm. A super cell approach was employed for both almost periodic and completely random structures. Nanorods are positioned within one super cell and the configuration repeats from one super cell to another by enforcing periodic boundary conditions in the in-plane directions.

Sunlight is incident on the structures from the top (red arrow). Incident light is modeled as a plane wave with a spectral distribution given by the ASTM Air Mass 1.5 direct solar irradiance spectrum. The optical constants of crystalline silicon are taken from Ref [14].

In order to quantify the broadband anti-reflection performance, we define the average reflection loss (ARL) as:

*λ*is wavelength,

*I*(

*λ*) is the solar irradiance spectrum, and

*R*(

_{TE}*λ*) and

*R*(

_{TM}*λ*) are the surface reflectance for TE and TM polarization, respectively. This figure of merit gives the ratio between the number of unabsorbed photons due to reflection loss and the total number of available photons above the band gap of crystalline silicon. The band gap corresponds to a wavelength

*λ*= 1127 nm. The solar spectrum is negligible below 310 nm. TE and TM polarized light have electric fields perpendicular or parallel to the incidence plane, respectively. We use a 3D full-vectorial scattering matrix solver (the ISU-TMM simulation package [15]) to calculate reflectance.

_{g}First, we evaluate the anti-reflection performance of periodic structures. The periodic structure is defined by a lattice constant *a* and wire diameter *d*, as shown in Fig. 2(a)
. We calculated the ARL as a function of *a* and *d/a*, the diameter to lattice constant ratio. The filling ratio of the array is related to the *d/a* ratio by *π*(*d/a*)^{2}/4. The calculation results are plotted in Fig. 2(b). Clearly, there is an optimal range of parameter space that minimizes broadband reflection. The optimal range for *d/a* is 0.6 to 0.8. In this range, lattice constants between 200 and 350 nm give the lowest ARL. A finer scan of the lattice constant and the *d/a* ratio near the optimal range reveals that the optimal periodic structure has an ARL of 4.64%, with lattice constant of 270 nm and *d/a* of 0.7 (equivalent to a filling ratio of 0.38).

The nanorod array can be viewed as an intermediate layer that improves the impedance matching between the air and the substrate. However, this effect depends on periodicity. In order to gain insight into the dependence of reflectance on lattice constants, we plot in Fig. 2(c) the reflectance spectra for periodic structures with lattice constants ranging from 100 nm to 1000 nm and a fixed filling ratio of 0.38 (corresponding to *d*/*a* = 0.7). The white dashed line indicates the diffraction limit in air (*a* = *λ*). Clearly, for broadband anti-reflection applications, the lattice constant should be kept below the diffraction limit for the lowest wavelength in the spectral range (310 nm in our case); otherwise, the diffuse (high order) reflectance will significantly increase the total reflection loss. On the other hand, below the diffraction limit, the optimal lattice constant is determined by the number and positions of reflection minima in Fig. 2(c). The positions of these minima are determined by the synergy between the propagating optical modes in the nanorod array, Fabry-Perot effects at the nanorod-air and the nanorod-substrate interfaces, as well as the opening of transmission orders. A detailed analysis of these spectral features might be carried out based on the modal method demonstrated in Refs. [16, 17], but is beyond the scope of this paper.

In Fig. 3
, we present a systematic comparison of reflection loss in periodic and various aperiodic structures. The filling ratio is fixed to 0.38, and the nanorod diameter is varied. For reference, the ARL values for a bare silicon substrate, for conventional pyramid surface texture, and for an optimized single-layer Si_{3}N_{4} anti-reflection coating with a thickness of 80 nm are shown by dashed lines. The reflectance spectrum of the pyramid surface texture was calculated by a simple scalar approach, following Ref. [18].

Consistent with Fig. 2, the ARL of the periodic structure (green line) decreases with increasing lattice constant until the optimal diameter of 189 nm (corresponding to an optimal lattice constant of 270 nm) is reached. The optimal periodic structure exhibits lower reflection loss than either the AR coating or pyramidal surface texturing.

Next, we determined the ARL for almost periodic structures. For each configuration, the rod positions were determined by adding a random shift to the original position of each rod, where the shift was drawn in each of the lateral directions from a uniform distribution with width equal to 10% of the lattice constant. The number of nanorods within one super cell was 25 for the smallest nanorod diameter (70 nm), and 16 for other structures. The super cell side length is equal to 500 nm, 600 nm, 800 nm, 1000 nm, and 1200 nm for nanorods with diameters of 70 nm, 105 nm, 140 nm, 175 nm, and 210 nm, respectively. A total of 100 configurations were calculated for each nanorod diameter. The symbols and error bars in Fig. 3 indicate the mean values and standard deviations of the ARL, respectively. The mean ARL lies slightly below the value for the periodic structure for smaller nanorod diameters, and slightly above for larger diameters. We conclude that overall, the anti-reflection performance of a periodic structure is not particularly sensitive to slight perturbations in position, for example, due to fabrication imperfections.

In contrast, completely random structures can exhibit very different reflectance from periodic structures. Random structures were generated by randomly positioning each rod within the super cell, subject only to the constraint that no two rods overlap. A total of 300 configurations were calculated for each nanorod diameter. For nanorod diameters below 150 nm, the mean ARL of the random structure is lower than that of the periodic structure. However, for larger rod diameters, the mean ARL of the random structure is higher than the periodic one. In short, the effect of aperiodicity on anti-reflection varies with feature size, and random structures are generally beneficial for small, non-optimal nanorod diameters.

The size-dependent role of aperiodicity can be explained in an intuitive manner. Randomization of nanorod positions within the super cell will introduce additional effective periodicities (spatial Fourier components). From Fig. 2(b), we observe at fixed filling ratio, the ARL decreases with increasing lattice constant and then increases again. For initial periodicities smaller than the optimal value (270 nm), the additional larger effective periodicities introduced by randomness tend to decrease the ARL. On the other hand, when the initial periodicity is larger than the optimal value, the even larger effective periodicities increase the ARL. We caution, however, that in the high index contrast structures studied here, the physical picture of reflection as resulting from a sum of gratings corresponding to the spatial Fourier components (effective periodicities) of the nanorod array [19, 20] may not strictly apply; further work is required to develop an appropriate scattering model.

Finally, we consider how deliberate design of an aperiodic structure can further improve antireflection performance. Starting from the periodic structure, we use a guided random-walk optimization algorithm similar to Ref. [21]. to obtain an aperiodic structure than minimizes the ARL. At each iteration, a nanorod is randomly moved to determine whether the ARL decreases; if so, the nanorod is kept in the new position. We ran 300 iterations for each nanorod diameter. We found that regardless of nanorod size, our algorithm found optimized structures with lower ARLs than the periodic structure. The ARL is also lower than the mean value of random structures. The exception is a nanorod diameter of 210 nm. In this case, the algorithm failed to identify an aperiodic structure with lower ARL than the starting periodic configuration. Optimized aperiodic structures with nanorod diameters between 70 nm and 175 nm all outperform the optimal periodic structure. Moreover, since random-search algorithms are not guaranteed to reach the global optimum, values of the ARL even lower than those in Fig. 3 (magenta line, “optimal”) may be possible with further optimization.

Figure 3 shows that the effects of disorder and aperiodicity cannot be described using simple effective medium theory. The nanostructured material is sometimes modeled with an effective index that depends only on the filling ratio and the refractive indices of silicon and air [22]. Reflection from the surface can then be calculated from a 1D problem, corresponding to a substrate coated with a homogeneous layer with the given effective index. The results of Fig. 3 show that structures with fixed filling ratio (and hence, the same effective index) can have radically different ARL values. Even for structures with fixed nanorod diameter, periodic, random, and optimal aperiodic structures have different values of ARL.

In order to gain insight into the role of aperiodicity in the anti-reflection performance, we plot in Fig. 4 the reflectance spectra of periodic, random, and optimized aperiodic structures for different nanorod diameters. For random structures, the reflectance spectra of 300 different configurations are shown. For a small, non-optimal nanorod diameter of 70 nm, the reflectance spectra of all random structures lie below that of the periodic one across most of the solar spectrum. The optimally designed aperiodic structure exhibits even lower reflectance. For a larger nanorod diameter of 140 nm, closer to the optimal value for periodic structures, the reflectance of the random structures can be lower, higher, or similar to that of the periodic structure, depending on wavelength. Nonetheless, the reflectance of the optimized aperiodic structure is still lower than the periodic one across the entire wavelength range. Eventually, when the nanorod diameter increases to 210 nm, random aperiodic structures exhibit higher reflectance than the periodic structure across the whole spectrum. The above observations are consistent with the results in Fig. 3.

We have investigated the dependence of ARL on incidence angle (zenith angle) for both periodic and random aperiodic structures. For each size, the ARLs for 50 random structures are calculated for incidence angles up to 80 degrees. The qualitative trends observed in Fig. 3 continue to hold up to large incidence angles.

In our study, the size of the super cell was chosen for the sake of computational feasibility. However, we have also calculated the ARL for 100 random structures at each nanorod diameter for four times the original supercell area. This set of simulations was done using the Lumerical FDTD simulation package. Similar to Fig. 3, random aperiodic configurations exhibit either lower or higher average values of ARL than the periodic structure, depending on nanorod diameter.

In summary, we have systematically investigated the effect of aperiodicity on the anti-reflection performance of silicon nanorod structures. Our results have important implications for practical device applications. First, for lithographically-defined periodic structures, fabrication errors in the position of the nanorods do not significantly degrade the anti-reflection performance. In fact, for small, non-optimal nanorod sizes, fabrication imperfections can slightly reduce the reflection loss. Secondly, for nanorod diameters below 150 nm, random structures generally outperform their periodic counterparts, suggesting that lithographic methods may not be necessary. Thirdly, for larger nanorod diameters, perfectly periodic structures are preferred, which requires effective lithographic methods to minimize the degree of structural randomness. However, for optimal performance, well-controlled, deliberately-designed aperiodic structures will give the best overall results.

We have carried out similar calculations for GaAs-based structures with the same height and similar dimensions. Although GaAs is about ten times more absorptive than silicon in the solar spectrum, a size-dependent effect of positional aperiodicity similar to the Si case (Fig. 3) was observed. On average, random structures outperform periodic ones for small nanorod diameters.

In previous work [21], we showed that, under certain conditions, aperiodic arrangements of silicon nanowire arrays in air (no substrate) absorb more light than periodic arrays. Other authors have reached similar conclusions for other types of aperiodic structures [23]. This observation motivated the current study. However, due to the fact that the physics is different for the two problems, the performance of the structures in Fig. 1 could not be predicted from previous results. The absorption in free-standing nanowire arrays depends on the degree to which the structure scatters the normally-incident light into in-plane optical modes. For nanorod surface textures (Fig. 1), we are not directly concerned with in-plane scattering, but only the extent to which the structure reduces reflection. Moreover, the reflection properties of the current system cannot be inferred from previous results and must be studied numerically using large-scale simulations.

Another related system consists of nanorod structures fabricated on thin substrates, or thin-film solar cells [10, 24]. We expect that harnessing structural randomness can significantly increase the device performance of thin film cells with nanostructured surface texturing. Further simulations are required to test this hypothesis.

## Acknowledgment

This work was funded by the Center for Energy Nanoscience, an Energy Frontiers Research Center funded by the U.S. Department of Energy Office of Science, Office of Basic Energy Sciences, under Award DE-SC0001013. Ningfeng Huang is funded by a USC Annenberg Fellowship. Computing resources were provided by the USC Center for High Performance Computing and Communications.

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