Leaky Mode Engineering: A General Design Principle for Dielectric Optical Antenna Solar Absorbers

Transcription

1 Leaky Mode Engineering: A General Design Principle for Dielectric Optical Antenna Solar Absorbers Yiling Yu, Linyou Cao, * Department of Materials Science and Engineering, North Carolina State University, Raleigh NC 7695; Department of Physics, North Carolina State University, Raleigh NC 7695; Abstract We present a general principle for the rational design of dielectric optical anatennas with optimal solar absorption: leaky mode engineering. This builds upon our previous study that demonstrates the solar absorption in a given amount of materials dictated by the modal properties of leaky modes. Here we synergistically examine the correlation among the modal properties of leaky modes, the physical features of dielectric structures, and the solar absorption in these structures. Our analysis clearly points out the general guideline for the design of dielectric optical antennas with optimal solar absorption enhenacement: a) using 0D structures; b) the shape does not matter much; c) heterostructuring with non-absorbing materials is a promising strategy; d) the design of a large-scale nanostructure array can literally build upon the design of single nanostructure solar absorbers. * To whom correspondence should be addressed. lcao@ncsu.edu

2 The strong, tunable optical resonance of subwavelength or wavelength-scale dielectric structures presents a promising strategy for the enhancement of solar absorption.[-4] The dielectric resonant structure can act as an optical antenna to efficiently trap incident solar light into a confined space. Unlike the surface plasmon resonance in metallic structures,[5-8] which may cause incident solar radiation lost as heat owing to the lossy nature of metal, dielectric optical antennas can ensure the absorption of solar radiation only occurring in the active materials of interest (semiconductor) and all converted into extractable charge carriers. Recent research have indeed demonstrated substantial solar absorption enhancements with a variety of dielectric structures, including nanoparticles, nanowires, nanopillars, nanocones, and nanowells.[9-6] Despite the extensive studies on dielectric optical antenna solar absorbers, there is no general principle available that can guide the rational design of antenna structures with optimal solar absorption enhancement. This is due to a limited understanding for the role of dielectric optical antennas in solar absorption. It is well recognized that the light trapping efficiency strongly depend on the physical feature, including dimensionality, morphology, structure, and composition, of the antenna.[9-9] But no intuitive yet quantitative understanding for the fundamental nature of these dependences has been established yet. Most of the existing research is case study, focusing on the absorption optimization of specific structures in a relatively narrow parameter window. The lack of general design guideline makes it very difficult to seek the ultimate optimization of solar absorption over a broad range of physical parameters, which is highly desired for the development of affordable solar electricity. Here we present a general principle for the rational design of dielectric optical antenna solar absorbers: leaky mode engineering. We demonstrate that key to optimize the solar absorption is to engineer the modal properties of leaky modes in dielectric optical antennas, more specifically, to increase the number of leaky modes and to tune the radiative loss of the leaky mode into a proper range. Leaky modes are natural optical modes with propagating waves outside the antenna structure, and each mode is featured with a complex eigenvalue (N real N imag.i) that can be solved analytically or numerically.[7-9] This study builds upon our previous work, which demonstrates the solar absorption in given materials dictated by the modal properties of leaky modes of the materials.[40] In this work, we systematically elucidate the correlation between the modal properties of leaky modes and the physical features (dimensionality, shape, etc.) of dielectric structures. We also explicitly illustrate that the difference in solar absorption of various dielectric structures can all be ascribed to the difference in modal properties of leaky modes. This work points out a powerful, general paradigm for the rational design of dielectric optical antenna solar absorbers from the perspective of leaky mode. We start with briefly reviewing the key discoveries of our previous studies.[40] We have demonstrated that semiconductor nanostructures of any dimensionality (zero, one, and two) can be considered as leaky resonators, and the light absorption of the nanostructure can be analyzed using an intuitive model, coupled leaky mode theory (CLMT).[7] The CLMT model considers the light absorption as coupling between incident light and the nanostructure s leaky modes (Figure a inset). For given semiconductor materials, the solar absorption of single leaky modes is found only dependent on two variables, the radiative loss q rad and the resonant wavelength λ 0, of the mode, regardless the physical feature of the materials. The two variables are related with the complex eigenvalue (N real N imag.i) of the leaky mode as q rad = N imag / N real and λ 0 = πnr/

3 N real, where n and r are the refractive index and characteristic size of the materials, respectively. We have also demonstrated that the solar absorption of single leaky modes with arbitrary resonant wavelength λ 0 can be maximized by tuning the radiative loss. As an example, Fig. a shows the calculated solar absorption of single leaky modes in D nanostructures as functions of the radiative loss q rad and the resonant wavelength λ 0. Amorphous silicon (a-si) is used as the absorbing material in the calculation.[4] The solar absorption is calculated by integrating the spectral absorption cross-section C abs (λ) of single leaky modes over the spectral photon flux of solar radiation I(λ) as " C abs (!)I(!)d!. We assume that all the absorbed photons can generate electrons with unity internal quantum efficiency, which gives the calculated solar absorption in D structures a unit of ma/cm. The white dashed line in Fig. a indicates the maximal solar absorption and associated radiative loss at every resonant wavelength. The absorption maximal and associated radiative loss is re-plotted as a function of λ 0 in Fig.b. For structures with multiple leaky modes, the total absorption is a simple sum of the absorption by each individual mode, P total = # P(! 0 )"(! 0 )d! 0, where P (λ 0 ) and ρ(λ 0 ) are the solar absorption of single leaky modes and the density of leaky modes at an arbitrary resonant wavelength λ 0, respectively (Fig. b). a Resonant wavelength (nm) (ma/cm) x Radiative loss q Resonant wavelength (nm) rad Figure. (a) Calculated solar absorption of single leaky modes in D structures as a function of radiative loss and resonant wavelength. a-si is used as the absorbing material. The white dashed line connects the absorption maximal at each resonant wavelength. The inset schematically illustrates the coupling of incident light with the leaky mode in nanostructures. (b) The absorption maximal (red line) and associated radiative loss (blue line) as a function of resonant wavelength. The shaded area schematically illustrates the integration of contributions from multiple leaky modes. b Solar absorption (ma/cm) x 0 4 ρ(λ 0 ) δλ Radiative loss q rad The deterministic correlation of solar absorption with leaky modes points out a general guideline for the rational design of dielectric optical antenna solar absorbers. First of all, as the solar absorption is a sum of the contribution from each leaky mode, the total number of leaky modes involved poses a fundamental limit for the solar absorption in a given amount of materials. Additionally, from Fig. we can find that to maximize the solar absorption requests the resonant wavelength and radiative loss of leaky modes to be tuned into certain optimal ranges. For a-si, these ranges are nm and for the resonant wavelength and the radiative loss, respectively (Fig. b). These optimal ranges can be understood from an intuitive perspective. The optimal wavelength range ( nm) is to match the spectral photon flux of solar radiation,

4 and can generally apply to all kinds of active materials for solar absorption, including Si, a-si, CdTe, CIGS, and organic materials. The optimal range for the radiative loss is to match the intrinsic absorption loss (defined as n imag /n real, n real, n imag are the real and imaginary part of the refractive index of the active materials, respectively) of the materials in the optimal wavelength range. The match of radiative loss and absorption loss for optimal solar absorption is analog to the critical coupling in waveguide-coupled resonators, which is known able to maximize the energy coupled into the resonator. [4] Therefore, the engineering of leaky modes poses as a general principle for the design of dielectric optical antennas to maximize the solar absorption in a given amount of active materials. More specifically, the engineering is to increase the number of leaky modes and to tune the leaky modes resonant wavelengths and radiative losses towards corresponding optimal ranges. Leaky modes can be engineered by control of the physical feature, such as dimensionality, shape, of dielectric structures. We have demonstrated that the modal properties (mode density, resonant wavelength, and radiative loss) of leaky modes strongly depend on physical features.[7, 8, 4] In the following, we explore to establish a more quantitative understanding of the correlation between the physical feature and the modal properties in the context of solar absorption. While in principle the physical feature may have an infinite amount of varieties, we focus on elucidating the correlation using a couple of representative structures. We also evaluate and compare the solar absorption in these structures. Unless specifically pointed out, all the solar absorptions in the following are calculated by integrating spectral absorption cross-sections C abs (λ) over the spectral photon flux I(λ) in solar radiation.[44] Of our interest is to optimize the solar absorption in unit area and/or unit volume of the active materials. Therefore, the comparisons of solar absorptions are all made between structures with comparable geometrical cross-section perpendicular to incident solar light (referred as incident cross-section) or comparable volume of absorbing materials.. Dimensionality: D vs. 0D structures The main effect of the dimensionality is on the number of leaky modes. We define the D and 0D structures as having one-dimensional and two-dimensional subwavenlength or wavelengthscale dimension in the direction perpendicular to the incident solar light. We have demonstrated that the number of leaky modes with resonant wavelength in a range of [λ, λ ] can be written as πan (/λ -/λ ) and 8πVn (/λ -/λ ) / for D and 0D structures, respectively, where A is the area of the D structure in the transverse direction and V is the volume of the 0D structure.[40] These expressions hold true for structures with arbitrary geometric shapes (i.e. triangular, rectangular, hexagonal, etc.). They may also reasonably apply to heterostructures that combine absorbing and non-absorbing materials, for instance, core-shell structures, in which A or V would be the area or volume of the absorbing materials. We can illustrate the effect of dimensionality by examining the solar absorption. Figure shows the calculated solar absorptions for a-si circular nanowires (D) and spherical nanoparticles (0D) as a function of radius r. To ensure that the solar absorptions is compared between the same volume of materials, the given result for the nanowire is the solar absorption by a segment in length of 4r/ of the nanowire. To get the given result, we first multiply the absorption crosssection of the nanowire (in unit of m) with a length of 4r/, and then integrate the resulting

5 product over the solar spectrum. We can find that the solar absorption of the 0D structure is generally much larger than that of the D structure, except at small radii, such as < 50 nm. The different solar absorption and its dependence on the radius are rooted in the number of leaky modes. The number of leaky modes in D structures in the wavelength of interest [400 nm, 800 nm] (πan (/λ -/λ )) is generally smaller than that of 0D structures (8πVn (/λ -/λ ) /) except when the radius is smaller than 9n (/λ -/λ )/[6(/λ -/λ )] 50 nm. We can further evidence the role of the number of leaky modes from the absorption spectra of D and 0D a-si nanostructures with the same radius (Fig. b). The 0D nanoparticle shows substantial larger absorption efficiency Q abs (Q abs is defined as the ratio of the absorption cross-section C abs with respect to the incident cross-section G, which is r and πr for the nanowire and the nanoparticle, respectively, Q abs = C abs /G). Using the method for the calculation of leaky modes we demonstrated previously, we can identify 0 leaky modes in the 0D nanoparticle, but only 9 leaky modes in the D nanowire in the given wavelength range, as indicated in Fig. b. In brief, with comparable volume and incident cross-section, 0D structures can provide higher solar absorption than D structures owing to a larger number of leaky modes. While the given results given are for spherical and circular structures, we confirm that this conclusion can generally apply to structures with arbitrary shapes, such as rectangular (Fig. c). a Solar absorption (ma) x 0 8 r 4r/ r Radius r (nm) b Absorption efficiency Qabs Wavelength (nm) Figure. (a) Solar absorptions of D a-si circular nanowires and 0D a-si spherical nanoparticles as a function of radius. (b) Spectral absorption efficiencies of the nanowire (blue curve) and the nanoparticle (red curve) in radius of 00 nm. Also plotted are the leaky modes in the nanowire (blue dots) and the nanoparticles (red dots). The given number indicates the degeneracy of each leaky mode. (c) Solar absorptions of D a-si square nanowires and 0D a-si cubic nanoparticles as a function of the size d. c x Solar absorption (ma) 4 d d Radius d (nm) d. Shape: Triangular, Rectangular, and Circular We also examine the leaky mode and solar absorption in nanostructures with different shapes. For the convenience of mode identification, we use D structures as an example, but the result can apply to 0D structures as well. We first examine structures with different shapes but the same volume and incident crosssection. Fig. a shows the solar absorption of a-si nanostructures with rectangular, spherical,

6 and triangular shapes. All these structures have the same incident cross-section as illustrated in Fig.a inset. The heights of the rectangular and triangular structures are set to make them have the same volume as the circular one. This is to ensure the same number of leaky modes in all the structures. We can find that all the structures exhibit similar solar absorptions with small (<5%) difference. The comparable solar absorption is rooted in comparable modal properties (resonant wavelength and radiative loss) of the leaky modes in these structures. Fig. b plots the modal properties of typical leaky modes in these structures. The horizontal axis is the real part of the eigenvalue N real, which is associated with the resonant wavelength λ 0 as λ 0 = nπd/n real, and the vertical axis is the radiative loss q rad. Whereas an individual leaky mode could bear certain difference among the different shapes, a large number of leaky modes, which is the typical case involved in solar absorption, show a similar distribution. The comparable modal properties can also be evidenced by comparable absorption efficiencies of these structures ( Fig. c). a x 0 b c Solar absorption (ma/cm) rectangular triangular circular πd/4 πd/ Size d (nm) d Radiative loss q rad rectangular triangular circular N real Wavelength (nm) Figure. (a) Solar absorption of D a-si structures with rectangular, triangular, and circular shapes as a function of the size d. All the structures have the same incident cross-section and the same volume of absorbing materials. (b) Modal properties of leaky modes in the rectangular, triangular, and circular structures. The horizontal and vertical axises are the real part of the eigenvalue and the radiative loss of leaky modes, respectively. (c) Spectral absorption efficiency of the rectangular, triangular, and circular structures with a size d of 00 nm. Absorption efficiency Qabs 0.5 rectangular triangular circular We further examine the rectangular and triangular a-si structures with different size ratios. Fig. 4a show the solar absorption of rectangular structures with the same volume of materials but different size ratios a:b (a and b are the height and the width of the structure, respectively) of /, 0.785,, and. As the different size ratios may result in different incident cross-sections, we normalize the solar absorption given in Fig. 4a with respect to the incident cross-section and plot the normalized solar absorption in Fig. 4b. We can find that the structure with larger incident cross-section has larger solar absorption but smaller normalized solar absorption. In another word, the structure with larger incident cross-section has larger solar absorption per volume of materials but smaller absorption per incident area. We have demonstrated that, from the standpoint of statistics, the modal properties of leaky modes only show trivial change within the given size ratio.[4] The substantial difference in the solar absorption can be mainly linked to the coupling directivity of incident solar light with leaky modes at different geometries.[45] A thorough discussion would be out of the scope of this work, but we can understand this difference from an intuitive perspective. Generally, a larger incident cross-section could facilitate the coupling of incident solar light with leaky modes due to a larger exposure to the incidence,

7 and gives rise to larger solar absorption per volume of materials (per mode). Meanwhile, a larger incident cross-section may result in a smaller height, which means a smaller number of leaky modes and hence smaller solar absorption per incident area. We can find a similar dependence on the size ratio in triangular structures (Fig. 4c-d). Solar absorption (ma/cm) a.5 a 0.5 Figure 4. (a) Total solar absorption and (b) normalized solar absorption of D a-si rectangular structures with different size ratios (a/b) of /, 0.785,, and, which are labeled as structure,,, 4, respectively. All the structures have the same volume of absorbing materials. The inset is a schematic illustration of the structure and incident geometry. (c) Total solar absorption and (d) normalized solar absorption of D a-si triangular structures with different size ratios (a/b) of 0.89, 0.866,.57,.87. All the structures have the same volume of absorbing materials. The inset is a schematic illustration of the structure and incident geometry. We can conclude that the shape of dielectric structures does not substantially affect the modal properties of leaky modes from a standpoint of statistics, although may generate minor effects. As a result, for a given amount of materials, the different shapes (circular, rectangular, triangular) with comparable incident cross-section all have similar solar absorption. And the shape with a larger incident cross-section (perpendicular to the incident direction of solar radiation) may have larger solar absorption per volume of materials, while the shape with a larger height (in the incident direction of solar radiation) can have larger solar absorption per incident area.. Heterostructures: core-shell and quasi-core-shell.5 x 0 c 4 b a 4 b crosssection area (nm ).5 Of our interest is the heterostructure consisting of absorbing and non-absorbing materials, such as a-si and ZnO. The non-absorbing materials do not absorb solar light by themselves, but may help enhance solar absorption in the absorbing materials. We have previously demonstrated that a-si (core)-zno(shell) heterostructures can substantially boost the solar absorption in the a-si core.[46] Other groups also report that heterostructures like void resonators, coated spheres/particles, may help substantially enhance the solar absorption.[6-9, 9, 0] We now b d Normalized solar absorption (ma/cm )

8 examine the improved absorption enhancement of heterostructures from the perspective of leaky modes, and demonstrate that it can be correlated to the effect of heterostructures on the radiative loss of leaky modes in the absorbing materials. We examine two types of heterostructures. As illustrated in Figure 5, in the type structure absorbing materials are partially or fully coated by nonabsorbing materials, and type has an opposite structure with nonabsorbing materials partially or fully coated by absorbing materials. Again, we use D structures as an example, but the results can apply to 0D structures as well. Type Heterostructure Type Heterostructure Absorbing materials Non-absorbing materials Figure 5. Schematic illustration for the heterostructures studied. Fig. 6a shows the calculated solar absorption for coated a-si structures as a function of the size of the a-si materials. The solar absorption of a pure a-si structure is also given as a reference. All the structures have the same volume of a-si materials. Without losing generality, the nonabsorbing coating is set to be in thickness of 70nm and with a constant refractive index of, which is close to that of ZnO. We can see the solar absorption of the fully and partially coated structures substantially larger than that of the pure structure. Fig.6b shows the spectral absorption efficiencies of the circular structures with a radius of 00 nm in the a-si part. This also indicates a substantial improvement in the absorption by the full and partial coatings. For visual convenience, the absorption spectrum of the rectangular heterostructure is not given. a x 0 b c Solar absorption (ma/cm) 0 4 r Radius r (nm) Wavelength (nm) Figure 6. (a) Solar absorption of pure (structure, black curve), partially coated (structure, red curve), fully coated (structure, blue curve), and coated rectangular (structure 4, orange curve) a-si structures as a function of the radius of the a-si part. The inset schematically illustrates the structures and incident geometry. (b) Spectral absorption efficiencies of the circular structures (structure,, ) with r = 00 nm. Also plotted are the leaky modes in the structures, fully coated (blue dots), partially coated (red dots), and pure (black dots). The size of the dot is proportional to the radiative loss of corresponding leaky mode. For visual convenience, the degeneracy of the leaky modes is not given. (c) The distribution of magnetic field intensity H z (z is along the longitudinal direction of the D structure) of a typical mode TE in the pure structure (top), fully coated (second top) and partially coated (the bottom two) structures. Absorption efficiency IHzI 0.5 0

9 We have previously demonstrated that the dielectric shell may act as an anti-reflection coating.[46] The role of the dielectric shell can be more generally understood from the perspective of leaky modes. We calculate the leaky modes involved in the absorption of these structures using analytical or numerical (COMSOL) techniques, and plot them as a function of resonant wavelengths in Fig. 5b. The size of each dot indicates the radiative loss of corresponding leaky mode (Fig. 5b). The number of leaky modes is largely conserved in these structures. We can find one-by-one correlation between the leaky modes in the coated structures and those in the pure a-si structure by examining the field distribution. Table lists the resonant wavelength, eigvenvalue, and radiative loss of typical leaky modes in these structures. It is worthwhile to note that by coating a dielectric shell may actually create new leaky modes, but we only consider the leaky modes evolved from the original modes in the a-si core, which typically show electromagnetic fields mainly distributed in the a-si materials and contribute the majority of absorption in the heterostructures. While the coating may cause certain change in the resonant wavelength of leaky modes (Table ), the improved absorption enhancement mainly results from the increase in the radiative loss. From Table we can find that all the leaky modes see increase in radiative loss after the coating, for instance, the radiative loss of the mode TM (the nomenclature of leaky modes can be seen in the note of Table ) changes from 0.04 in the pure structure to around 0.04 in the coated structures. We can know from Fig. that the increase in radiative loss towards the optimal range of can make the coupling of leaky modes with incident solar light more efficient, which can result in larger solar absorption. The change in radiative loss can be more clearly seen from the eigenfield distribution of leaky modes. Fig. 6c show the distribution of magnetic field intensity of a typical leaky mode TE in the pure and coated a-si structures. We can find that the field in the coated structure is obviously more spread than that in the pure structure, indicating a larger radiative loss. Correspondingly, the radiative loss of this mode changes by one order of magnitude from in the pure structure to 0.06 or 0.05 and 0.07 in the half-coated and fully-coated structures, respectively.

10 Table. Leaky modes in pure, partially and fully coated a-si nanostructures Pure Partially coated Fully coated! 0 (nm) Eigenvalue q! 0 (nm) Eigenvalue q rad! 0 (nm) Eigenvalue q rad TM i i i i 0.05 TM i i Ii 0.0 TM i i i i 0.0 TM i i i i 0.06 TM i i i i 0.00 TE i i i i 0. TE i i i i 0.06 TE i i i 0.08 TE i i i i 0.07 TE N/A i i N/A i N/A i 0.08 Note:. TE and TM refer to transverse electric and transverse magnetic polarizations, respectively. The two subscript numbers indicate the mode number and the order number of the leaky mode.[7, 8]. Most of the given modes in the pure and the fully coated structures have a dual degeneracy except the TM 0 and TE 0 modes. But the dual degeneracy disappears in the partially coated structure. We can find similar absorption enhancement improvement in the type structure, and can also correlate it to the increase in radiative loss of leaky modes. Fig.7a shows the solar absorption of the full and partial a-si coatings as a function of size. Without losing generality, for the circular heterostructures, we set the radius of the core and the thickness of the shell in ratio of 4:, and set the refractive index of the core to be.5, which is close to that of SiO. The rectangular heterostructure is set to be a rectangular core coated by a layer of a-si materials with similar dimensions as the counterparts in the circular structures. Although the a-si coatings have a much smaller volume of active materials, their solar absorptions are close to that of the pure a-si structure. We can get more intuitive understanding from the absorption spectra of the coatings and the pure structure (Fig. 7b). For simplicity, only the spectra of the circular structures are given. We can find that the coatings may have comparable or even larger absorption than that of the pure structure in certain wavelength ranges, for instance, from 580 nm to 60 nm (Fig. 7b). This can be correlated to the increase of the radiative loss of leaky modes. Fig. 7c plots the leaky modes of these structures with resonant wavelength in the range of nm. The number of leaky modes in the pure structure (black dots) is apparently larger than the full (blue dots) and partial (red dots) coatings. However, we can find multiple leaky modes in the coatings with radiative loss substantially larger than those in the pure structure. The gain resulting from the increase of radiative loss may substantially offset the loss caused by the decrease in the number of leaky modes.

11 a x 0 b c Solar absorption (ma/cm) πr/ r Radius r (nm) Wavelength (nm) Figure 7. (a) Solar absorption of pure (structure, black curve), partial coating (structure, red curve), ful coating (structure, blue curve), and rectangular partial coating (structure 4, orange curve) a-si structures as a function of the size. The inset schematically illustrates the structures and incident geometry. The radius of the core and the thickness of the shell is set to be in a ratio of 4:. The refractive index of the core is.5. (b) Spectral absorption efficiencies of the circular structures (structure,, and ) with r = 75 nm. (c) The modal properties of leaky modes in the structures with resonant wavelengths in the range of nm, full coating (blue dots), partial coating (red dots), and pure (black dots). For visual convenience, the degeneracy of the leaky modes is not given. Generally, most of the leaky modes in the pure structure and the full coating have a degeneracy of two, but the mode in the partial coating does not have the degeneracy. Our analysis indicates that heterostructuring poses a promising strategy for the engineering of leaky modes. The radiative loss of leaky modes in the absorbing materials may be substantially increased by the heterostructuring. This is in essence because the heterostructuring changes the refractive index contrast of the absorbing materials. The effect of the heterostructuring on the readiative loss of leaky modes accounts for all the observed improvements in the solar absorption of heterostructures reported in literature. 4. Single nanostructures and an array of nanostructures Absorption efficiency Qabs 0.5 A large scale array of nanostructures is the most interesting for practical applications. However, the rational design of single nanostructure absorbers would be much less time-consuming. Therefore, understanding the correlation between the solar absorption of single nanostructures and that of an array of nanostructures can significantly facilitate the rational design of solar absorbers for practical applications. We use single D structures and a periodic array of D structures as an example to address this issue from the perspective of leaky modes. We find that, with a reasonable inter-nanostructure spacing (typically not too small compared to the size of each nanostructures), the leaky modes in a periodic array of nanostructures are similar to those in single nanostructures. Fig. 8 shows the absorption spectra of single a-si D structures in radius of 00nm and an array of the same nanostructure with 400 nm in the inter-nanostructure distance (Fig.8 inset). For a fair comparison, the absorption efficiency of the array is calculated with respect to the entire space including the opening between nanostructures, and that of the single nanostructure is calculated by normalizing its absorption cross-section with respect to a size of 400 nm. We can find that the absorption spectra of both structures are reasonably similar. To understand the absorption similarity, we calculate the leaky modes of the single nanostructure and the array using analytical Radiative loss N real

12 and numerical (COMSOL) techniques, and plot the leaky modes along with the absorption spectra. We also list the detailed modal properties of the leaky modes in the table of Fig. 8. For most of the leaky modes in the nanostructure array, we can find a one-by-one correlation with the mode in the single nanostructure by examining the field distribution. Literally, in the absorption spectra given in Fig.8, only one leaky mode in the nanostructure array cannot be correlated to any mode in the single structure (marked in blue color in the figure and the table). The new mode can be recognized resulting from the Bragg scattering in the nanostructure array. This result indicates that a predominant majority of the leaky modes in the nanostructure array originate from the mode of single nanostructures, and that the number of new modes resulting from the collective optical response of the array is trivial. From the table given in Fig. 8, we can also find that these correlated modes show similar radiative losses and resonant wavelengths. As a result of the similar number of modes and modal properties, each individual nanostructure in the array may absorb solar radiation in a similar way as the single nanostructure. This conclusion holds for any nanostructure array with interspacing not too small comparable to the size of individual nanostructures. Absorption efficiency 0.5 4r r Wavelength (nm) Single Array 0(nm) Eigenvalue q rad 0 (nm) Eigenvalue q rad TM i i 0.4 TM i i 0.0 TM i i 0. TM i i 0.00 TM i i TM i i TE i i TE i i 0.04 TE i i 0.05 TE i i TE i i i 0.0 Figure 8. (Left) Spectral absorption efficiencies of single a-si D structures with a radius of 00 nm and of an array of the structure with 400nm in the interspacing. Also plotted are the leaky modes in the single structure (black dots) and the nanostructure array (red dots). The blue dot indicates the leaky mode in the nanostructure array that cannot be correlated to any mode in the single structure. (Right) tabulated modal properties for the leaky modes in the single structure and the nanostructure array. Again, the blue color indicates the leaky mode in the nanostructure array that cannot be correlated to any mode in the single structure. The similarity in the leaky modes of single nanostructures and an array of nanostructures confirms that the solar absorption in the individual nanostructure of the array can be reasonably believed similar to that of single nanostructure. This conclusion is very useful for the rational design of solar absorbers in practical applications. We can first focus on design single nanostructures with optimal solar absorption, and then use the designed structure as building blocks to build up a large scale array. As the final note, leaky mode engineering represents the general principle for the rational design of dielectric optical antennas with optimal solar absorption enhancement. Results of this work

13 further points out the design guideline, a) using 0D structures; b) the shape does not matter much; c) heterostructuring with non-absorbing materials is a promising strategy; d) the design of a large-scale nanostructure array can literally build upon the design of single nanostructure solar absorbers. It is worthwhile to note that more studies would be necessary to find out better heterostructures to further engineer the radiative loss of leaky modes. For instance, the tunability in the radiative loss of leaky modes using the given heterostructures is limited (< 0.), which is still substantially below the optimal range of a-si materials. Acknowledgements This work is supported by start-up fund from North Carolina State University. L. Cao acknowledges a Ralph E. Power Junior Faculty Enhancement Award for Oak Ridge Associated Universities References. C. F. Bohren, and D. R. Huffman, Absorption and Scattering of Light by Small Particles (John Wiley & Sons, Inc., New York, 998).. L. Y. Cao, J. S. White, J. S. Park, J. A. Schuller, B. M. Clemens, and M. L. Brongersma, "Engineering light absorption in semiconductor nanowire devices," Nature Materials 8, (009).. Z. F. Yu, A. Raman, and S. H. Fan, "Nanophotonic light- trapping theory for solar cells," Appl Phys a- Mater 05, 9-9 (0). 4. A. Polman, and H. A. Atwater, "Photonic design principles for ultrahigh- efficiency photovoltaics," Nature Materials, (0). 5. H. A. Atwater, and A. Polman, "Plasmonics for improved photovoltais devices," Nature Materials 9, 05- (009). 6. R. A. Pala, J. White, E. Barnard, J. Liu, and M. L. Brongersma, "Design of Plasmonic Thin- Film Solar Cells with Broadband Absorption Enhancements," Advanced Materials, (009). 7. J. A. Schuller, E. S. Barnard, W. Cai, Y. C. Jun, J. S. White, and M. L. Brongersma, "Plasmonics for extreme light concentration and manipulation," 9, 9-04 (00). 8. K. Aydin, V. E. Ferry, R. M. Briggs, and H. A. Atwater, "Broadband polarization- independent resonant light absorption using ultrathin plasmonic super absorbers," Nature Communications (0). 9. L. Cao, P. Fan, A. P. Vasudev, J. S. White, Z. Yu, W. Cai, J. A. Schuller, S. Fan, and M. L. Brongersma, "Semiconductor Nanowire Optical Antenna Solar Absorbers," Nano Letters 0, (00).

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