## Abstract

We design silicon membranes with nanohole structures with optimized complex unit cells that maximize broadband absorption. We fabricate the optimized design and measure the optical absorption. We demonstrate an experimental broadband absorption about 3.5 times higher than an equally-thick thin film.

© 2013 OSA

## 1. Introduction

Nanophotonic light-trapping structures [1, 2] promise to reduce the material cost and improve the efficiency of solar cells. Most light-trapping structures investigated so far have consisted of periodic arrays of sub-wavelength features. Recently, however, alternative structures have been proposed based on aperiodic patterns [3–5]. Aperiodic structures were found to improve solar absorption, with the additional benefit of insensitivity to incidence angle and polarization. However, their design and optimization are a severe challenge, due to the difficulty of simulating the electromagnetic fields of a truly aperiodic structure. Computational work has instead focused on structures that are aperiodic on smaller length scales, while periodic on a larger scale [6–15] (typically on the order of a micron). The large-scale periodicity provides a finite computational cell for calculations, making such structures amenable to simulation and optimization. Theoretically, it has been predicted that structures with optimized unit cells show strong broadband absorption enhancement [8, 10, 13, 15]. However, direct comparison with experiment has been limited.

In this work, we design structures with optimized complex unit cells in a silicon membrane that maximizes absorption over the solar spectrum. We then fabricate the optimal structure and measure the absorption in the 600 – 1000 nm wavelength range. The optimal structure absorbs about 3.5 times more light than a thin film of equal thickness, using only 76% silicon volume. Our results show excellent agreement between theory and experiment, suggesting the broad applicability of optimization techniques for photovoltaic structure design.

## 2. Design and optimization

Figures 1(a) and 1(b) illustrate the silicon nanohole
structures under study. We consider a free-standing silicon nanomembrane with thickness
*t* = 310 nm patterned with holes of diameter *d* = 120 nm. The
unit cell is a square with side length *L* and repeats in both the
*x* and *y* direction. The boundaries between the unit cells are
indicated by the white dashed lines. Within each unit cell, we position (*L*/200
nm)^{2} holes in various fashion, including a simple periodic pattern (Fig. 1(a)) and complex aperiodic geometries (Fig. 1(b)). The silicon filling ratio is 71.7% for all
structures.

We carried out systematic numerical simulations to predict the effect of complex unit cell geometries on broadband absorption. We use the scattering matrix method (ISU-TMM) [16] to calculate the absorptance spectra of various structures at normal incidence. Absorptance spectra for two orthogonal polarizations were averaged to obtain the average absorptance for randomly polarized incident light. We use the tabulated optical constants for crystalline silicon [17] for our simulations. In order to quantify the broadband optical absorption performance in the solar spectrum, we define a figure of merit (F.O.M.) as follows:

*λ*is the wavelength,

*I*(

*λ*) is the ASTM Air Mass 1.5 direct and circumsolar solar irradiance spectrum [18], and

*A*(

*λ*) is the absorptance spectrum of the structure averaged for two orthogonal polarizations (s and p) at normal incidence. This F.O.M. represents the ratio between the number of absorbed photons and the total number of incident photons in the incident solar spectrum;

*λ*

_{min}and

*λ*

_{max}define the wavelength range of interest and were taken to be 400 nm and 1100 nm, respectively.

We first investigated structures with randomly positioned holes inside a unit cell, or random
complex unit cells. For each unit cell length, 100 different structures were generated by
placing holes at random positions within the unit cell. A minimum inter-hole distance of 40 nm
was enforced, due to fabrication constraints. The mean value and standard deviation of the
F.O.M. over the 100 structures are indicated by blue squares and error bars in Fig. 1(c), respectively. The F.O.M.s for the simple periodic
unit cell (*a* = 200 nm, *d* = 120 nm) and an un-patterned silicon
thin film of equal thickness are also given for reference. On average, structures with random
complex unit cells outperform their simple periodic counterpart, regardless of the unit cell
size.

We next optimized the hole positions to maximize the broadband absorption. For each unit cell length, we used a random walk optimization technique [8] to maximize the F.O.M.; starting from the simple periodic unit cell, 100 updates of the algorithm were performed. Our simulation results are shown by the red star symbols in Fig. 1(c). Structures with optimized complex unit cell geometries have significantly higher F.O.M.s than the mean values for structures with random complex unit cells. The optimized structure with the highest F.O.M. has a unit cell length of 600 nm.

In a separate simulation (not shown), we optimized the lattice constant of a periodic nanohole
array with the same filling ratio as the structures above. The periodic nanohole array has only
one nanohole in the center of each unit cell. The highest F.O.M. (28.35%) was obtained for a
lattice constant of 600 nm (hole diameter *d* = 360 nm). The structure with the
optimized unit cell geometry in Fig. 1(c) significantly
outperforms this result, possibly by breaking the symmetry within the unit cell and increasing
the number of resonances that can couple to external radiation [1, 15, 19–21].

## 3. Fabrication and measurement

Drawing on the above analysis, we focus on structures with a unit cell length of 600 nm. The
patterns chosen for fabrication are shown in Figs.
2(a)–2(c). Figure 2(a) illustrates the simple periodic unit cell. Figure 2(b) is a structure with a F.O.M. closest to the mean value of our 100 randomly
generated configurations. We refer to this configuration as the *average complex*
structure. Figure 2(c) shows the structure with the
*optimized complex* unit cell geometry.

We use electron beam lithography combined with inductively coupled plasma reactive ion etching (ICP-RIE) to transfer the nanohole patterns into the device layer of a silicon-on-insulator (SOI) wafer (SOITEC). Every pattern was defined within a circular region with a diameter of 50 µm. After the pattern formation, a 6 mm by 6 mm silicon membrane area was defined by standard UV photolithography and ICP-RIE. The center square part of the silicon membrane with dimensions of 500 µm by 500 µm was aligned to overlap the nanohole patterns, while the rest of the membrane was patterned with access holes to facilitate the wet etching of the buried oxide (BOX) layer. Finally, the silicon membrane was released from the handle silicon wafer in 49% hydrofluoric acid (HF) and wet-transferred [22] to an oxidized silicon wafer. The device area was centered over a perforated window in the carrier wafer to obtain a free-standing membrane [23], so as to yield accurate measurements of absorption in the silicon layer. Figures 2(d)–2(f) show scanning electron microscope (SEM) images of fabricated structures with simple periodic (d), average complex (e), and optimized complex (f) unit cell geometries. Image processing software [24] revealed that the differences in silicon filling ratio between these patterns are below 2%.

We characterized the optical absorption in the fabricated samples in a customized white light
spectroscopy setup consisting of a tungsten lamp and a fiber-coupled spectrometer (Ocean Optics
USB 4000). The collimated, unpolarized incident light was focused by a microscope objective
(10x, N.A. = 0.26) to only illuminate the patterned area of interest (circular area with 50
µm diameter). The reflected light from the sample was collected by the same objective,
while the transmitted light was collected by an achromatic doublet lens (*f* =
30mm, N.A. = 0.39). The collected light was focused into multimode fibers connected to the
spectrometer for analyzing both the reflectance
*R*_{exp}(*ω*) and transmittance
*T*_{exp}(*ω*) spectra. The experimental
absorptance spectrum *A*_{exp}(*ω*) can then be
determined by *A*_{exp}(*ω*) = 1 -
*R*_{exp}(*ω*) -
*T*_{exp}(*ω*). We used a silver mirror as the
reflection reference while the transmission reference was air. We focus on the wavelength range
from 600 nm to 1000 nm, in order to minimize the diffractive reflection and transmission, which
can escape the collection optics and lead to measurement errors (please see the
*Appendix* section for details). For a unit cell length of 600 nm, wavelengths
above 600 nm are prohibited from diffracting to air for a normally-incident plane wave. For a
weakly focused incidence beam as in our current experimental configuration, diffraction happens
above 600 nm. However, we have carefully examined this issue using standard diffraction theory
and numerical simulations (please see the *Appendix* section for details). We
have verified that the difference caused by diffraction between the actual and measured
broadband absorptance is negligible (less than 2% relative difference in terms of the
F.O.M.).

Our experimental results are shown in Figs. 3(a)–3(c). Figure 3(a) shows the reflectance. The thin film (gray) exhibits characteristic Fabry-Perot fringes. The structure with a simple periodic unit cell (red) also exhibits Fabry-Perot fringes, with an overall spectral shift to lower wavelengths, corresponding to a reduced effective refractive index. The structure with the average complex unit cell (blue) has a similar overall shape to the periodic structure, but with notable peaks and dips in the spectra. These features are characteristic of guided resonance modes [25, 26]. The structure with the optimized complex unit cell (green) has a much higher number of the guided resonance features. Figure 3(b) shows the transmittance. Fabry-Perot and guided resonance features are again visible in the spectra. The optimized structure has a significantly lower transmittance than the other structures. Figure 3(c) shows the absorptance. The F.O.M.s in the range between 600 nm and 1000 nm for the three structures are indicated by the numbers on the plot. It can clearly be seen that the optimized structure has the highest F.O.M. (29.60%), significantly higher than the average complex unit cell (16.85%) and 4.3 times higher than the simple periodic unit cell (5.60%). Thus, merely by designing the geometry within the unit cell, dramatic absorption enhancement occurs. We note also that the optimized structure has 3.5 times higher F.O.M. than a solid film with the same thickness (6.56%), with only 76% of the silicon volume.

## 4. Comparison to simulation and discussion

We simulate the fabricated structures assuming a silicon membrane thickness of 328 nm and a
*uniform* nanohole diameter of 110 nm. The thickness was determined by fitting
the measured optical transmittance spectrum of an un-patterned area, and the nanohole size was
found to give the best agreement between simulation and measurement. The results are shown in
Figs. 3(d)–3(f). Overall, we observe excellent agreement between simulation and measurement. The
simulated spectra have somewhat sharper features than the experimental spectra. This is possibly
due to the fact that non-uniformity in the size and shape of nanoholes in the fabricated sample
may broaden the spectral features in experiment [27].
Another factor is the finite spread of angles in the incident beam. In TMM simulations, we have
observed that averaging the spectra over a range of incidence angles yields smoother and broader
spectral features. In addition, the simulated F.O.M.s between 600 nm and 1000 nm for the three
different structures are indicated by numbers on Fig.
3(f). In comparison with the measured F.O.M.s, relatively good agreement was obtained.
The largest difference occurs for the optimized complex unit cell geometry, possibly due to the
high sensitivity of the optimized geometry to fabrication imperfections.

In order to qualitatively explain why the optimized complex unit cell geometry gives higher absorption than other unit cell geometries, we examine the spatial Fourier spectra. Figures 4(a)–4(c) again show the real space dielectric functions for reference. Figures 4(d)–4(f) show the corresponding spatial Fourier transforms of the dielectric function; the amplitudes are normalized to the DC component. The Fourier transform of the simple periodic unit cell is a square lattice, with discrete, non-zero components. The Fourier transform of the average complex unit cell is more uniform than the simple periodic configuration. The Fourier transform of the optimized complex unit cell has larger nearest-zero Fourier components than the average complex unit cell ($({k}_{x},{k}_{y})=\pm (2\pi /600,0)n{m}^{-1}$or $\pm (0,2\pi /600n{m}^{-1})$); these are the Fourier components corresponding to the unit cell periodicity.

We can relate the spatial Fourier transforms to the guided resonance modes of the structure. For a periodic structure, there is an onset frequency for guided resonance modes that scales linearly with the lattice constant [26]. Conversely, in terms of wavelength, a structure will support guided resonance modes up to a maximum wavelength that increases linearly with lattice constant. For the structure in Fig. 4(a), the periodicity is 200 nm, and guided modes are visible in the spectrum at about 625 nm and 680 nm (Fig. 3(f)), but not above. The average complex unit cell geometry in Fig. 4(b) has nonzero Fourier components at smaller wave vectors than the simple periodic unit cell ($\left|{k}_{x}\right|$and $\left|{k}_{y}\right|<3\times (2\pi /600)n{m}^{-1}$.) corresponding to effective lattice constants between 200 and 600 nm. Such Fourier components increase the maximum wavelength for guided resonance modes: from Fig. 3(f), it can be seen that guided resonance features appear up to wavelengths of 1000 nm, significantly increasing the broadband absorption. The optimized complex unit cell geometry also gives guided resonance features up to 1000 nm, and the overall absorption is the highest. The broadband absorption depends upon the number of resonances, the spacing between resonances, and the external coupling rate for each resonance [1, 28, 29]. The optimization procedure tends to select structures with largest nearest-zero Fourier components, implying that these components correlate with the optimal resonance conditions.

We have also calculated the angular response of the nanohole structure with an optimized
complex unit cell. Results for incident light with two orthogonal polarizations (s and p) and
their average are shown in Fig. 5. We found that it outperforms an un-patterned thin film with equal thickness (310 nm)
regardless of the incidence angle. Moreover, the polarization-averaged F.O.M. for the optimized
complex unit cell is higher than the optimized simple periodic unit cell (*a* =
600 nm, *d* = 360 nm) at all angles. The better angular response for p-polarized
incident light (Fig. 5(a)) for all structures can be
attributed to the lower reflection for p polarization at large incident angles. Finally, the
average F.O.M. for the optimized complex unit cell can remain above 80% of the normal incidence
value for an incidence angle as large as 70 degrees.

## 5. Conclusions

In summary, we demonstrate that silicon nanohole membranes with complex unit cell geometries can yield higher broadband optical absorption than an equally-thick silicon film. In particular, an optimized complex unit cell geometry gives a significantly better performance (3.5 times in the current study) than an un-patterned silicon slab, using only 76% of the absorbing material volume.

Given the large parameter space of the optimization problem, our results may be understood as a lower limit on the obtainable performance: further optimization may yield complex structures with even higher broadband absorption. Reverse design approaches based on ideal Fourier space distributions [11] and/or optimal resonant coupling conditions [28] are interesting directions for future research.

## Appendix

In our measurement setup, we use microscope objectives with a finite numerical aperture to
collect the light reflected from and transmitted through the patterned silicon membrane at
normal incidence. Therefore, the specular component of the reflection and transmission will
always be completely collected. However, diffractive reflection and transmission, which exist
for wavelengths below the diffraction threshold (see below), will likely escape the collection
optics and give an *underestimation* of the total reflectance *R*
or transmittance *T.* As a result, the absorptance *A* will be
*overestimated*, since *A* is determined from *A*
= 1 – *R* – *T.* In this section, we carefully
examine the diffraction phenomenon and quantify its effect on the measured absorptance under
current measurement conditions. Our simulation results predict a relative difference of less
than 2% between the measured and actual absorption, in terms of the F.O.M. in the chosen
wavelength range of interest (600 nm to 1000 nm). This difference is much less than the
difference in F.O.M. for different unit cell geometries.

We start by looking at a simple situation, where light with an incidence angle
*θ _{i}* strikes a one-dimensional periodically-patterned slab
with a lattice constant

*a*, as shown in Fig. 6. The incident light will be specularly reflected and transmitted with an angle equal to the incidence angle

*θ*, as dictated by Snell’s law. In addition, when the frequency of the incident light is high enough, discrete diffractive reflection and transmission orders also occur, as shown by the green dashed arrows in Fig. 6. The diffraction angle

_{i}*θ*can be determined from the conservation of photon energy and wave vector parallel to the slab interface (with the addition of multiple reciprocal lattice vectors) [30]. In particular,

_{d}*θ*= asin$\left(\frac{{k}_{}+2\pi m/a}{{k}_{0}}\right)$ = asin$\left(\frac{{n}_{i}\omega \mathrm{sin}{\theta}_{i}/c+2\pi m/a}{{n}_{i}\omega /c}\right)$, where

_{d}*n*is the refractive index of the incidence and exit medium,

_{i}*θ*is the incidence angle, and

_{i}*m*is an integer indicating the diffraction order. It can be shown by setting

*θ*= −90 degrees and

_{d}*m*= −1 that the threshold wavelength

*λ*at which the first order diffractive reflection and transmission appears is given by

_{th}*a/λ*= $\left(\frac{1}{{n}_{i}(1+\mathrm{sin}{\theta}_{i})}\right)$. For wavelengths smaller than

_{th}*λ*|

_{th},*θ*| < 90 degrees and diffractive reflection and transmission will occur.

_{d}According to the diffraction threshold condition given above, for a perfect plane wave at
normal incidence (*θ _{i} =* 0°) in air
(

*n*= 1), the diffraction threshold condition is reduced to

_{i}*a/λ*1, which means diffraction is forbidden for wavelengths larger than the lattice constant (

_{th}=*λ*>

*λ*=

_{th}*a,*or

*a/λ <*1). It can easily be shown that the diffraction threshold condition for a two dimensional square lattice array, as in the silicon nanohole structure, is the same as the one-dimensional case, where

*a*is now the lattice constant of the 2D square lattice. Therefore, in our nanohole structure, for wavelengths above 600 nm (the unit cell length), no diffraction occurs for a plane wave at normal incidence.

In practice, however, we use a weakly-focused incidence beam that inevitably has a finite
angular spread. We can estimate the angular spread using the effective focal length
*f* of the focusing microscope objective (20 mm for our 10x objective), as well
as the beam spot diameter *d* at the entrance pupil of the objective (measured
to be 4 mm). The half angle of the incident cone of light is determined from
*θ*_{max} = atan(*d*/2*f*), which
is approximately 5.7 degrees. With this angular spread, the diffraction threshold condition
gives a threshold wavelength *λ _{th}* of around 660 nm, and
diffraction occurs between 600 nm and 660 nm.

In order to quantify the diffraction loss, we performed scattering matrix method simulations
(ISU-TMM) to calculate the specular (zero^{th} order) and diffractive
*(*non-zero^{th} orders*)* reflectance and transmittance
spectra for the three different unit cell geometries under study, for polar incidence angles up
to 10 degrees. The total reflectance and transmittance spectra can be readily obtained by
summing up the specular and diffractive components. Assuming a top-hat angular power
distribution [27], we can obtain the reflectance and
transmittance spectra averaged over incidence angles up to
*θ*_{max}, the half angle of the incidence cone of light. We can
then determine the incidence angle-averaged “actual” and “specular”
absorptance from the total and specular-only reflectance and transmittance, respectively. The
specular spectra predict our measurement results.

Calculation results for the optimized complex unit cell are shown in Fig. 7(a). As expected, for a normally-incident plane wave
(*θ*_{max} = 0 degree), the actual absorptance spectra for
wavelength above the unit cell length (600 nm, indicated by the dashed line) are exactly the
same as the specular one. As the angular spread of the incidence beam increases (increasing
*θ*_{max}), the wavelength at which the specular and actual
absorptance coincide shifts upward, as indicated by the blue arrows. We have confirmed that
this wavelength is determined by the diffraction threshold condition in air, given by
*a/λ _{th}* = $\left(\frac{1}{1+\mathrm{sin}{\theta}_{i}}\right)$. Moreover, we observe that the difference between the actual and
specular absorptance spectra above 600 nm is small.

In order to estimate the effect of diffraction on the measured broadband absorption, we
evaluate the F.O.M.s from the actual and specular absorption spectra. The F.O.M.s evaluated for
wavelengths between 600 nm and 1000 nm are shown in Fig.
7(b). We observe that for the simple periodic unit cell, the specular F.O.M. is equal
to the actual one for an angular spread up to 10 degrees. This is due to the absence of
diffraction for the small periodicity (*a* = 200 nm). For the average and
optimized complex unit cells, the specular F.O.M. is equal to the actual one for a zero angular
spread, corresponding to a perfect plane wave. For a non-zero angular spread, corresponding to
a focused incidence beam, the specular F.O.M. is higher than the actual one due to diffraction
loss. Nevertheless, even for the optimized complex unit cell, which has the largest discrepancy
between the specular and actual F.O.M., the difference in F.O.M. is only 0.42% (absolute value)
for an angular spread of 6 degrees. This difference corresponds to the error in the absorption
measurement. However, the error is far less than the difference in F.O.M. between the three
unit cells (simple periodic, average complex, and optimized complex), and is therefore
negligible.

## Acknowledgments

The authors thank Jing Ma for assistance in microfabrication and Ningfeng Huang, Apoorva Athavale, Bostjan Kaluza, and Milind Tambe for discussions of features of optimal aperiodic structures. Chenxi Lin was supported by the USC Graduate School’s Theodore & Wen-Hui Chen Fellowship. Materials, supplies, and partial summer salary support for Michelle Povinelli were 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 No. DE-SC0001013. Development of the fabrication process for silicon nanohole slabs (Luis Javier Martínez) was funded by the Army Research Office under Award No. 56801-MS-PCS. Computing resources were provided by the USC Center for High Performance Computing and Communication (HPCC).

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