## Near-unity broadband absorption designs for semiconducting nanowire arrays via localized radial mode excitation |

Optics Express, Vol. 22, Issue S3, pp. A930-A940 (2014)

http://dx.doi.org/10.1364/OE.22.00A930

Acrobat PDF (3152 KB)

### Abstract

We report design methods for achieving near-unity broadband light absorption in sparse nanowire arrays, illustrated by results for visible absorption in GaAs nanowires on Si substrates. Sparse (<5% fill fraction) nanowire arrays achieve near unity absorption at wire resonant wavelengths due to coupling into ‘leaky’ radial waveguide modes of individual wires and wire-wire scattering processes. From a detailed conceptual development of radial mode resonant absorption, we demonstrate two specific geometric design approaches to achieve near unity broadband light absorption in sparse nanowire arrays: (i) introducing multiple wire radii within a small unit cell array to increase the number of resonant wavelengths, yielding a 15% absorption enhancement relative to a uniform nanowire array and (ii) tapering of nanowires to introduce a continuum of diameters and thus resonant wavelengths excited within a single wire, yielding an 18% absorption enhancement over a uniform nanowire array.

© 2014 Optical Society of America

## 1. Introduction & Motivation

1. L. Hu and G. Chen, “Analysis of optical absorption in silicon nanowire arrays for photovoltaic applications,” Nano Lett. **7**(11), 3249–3252 (2007). [CrossRef] [PubMed]

3. P. Spinelli, M. A. Verschuuren, and A. Polman, “Broadband omnidirectional antireflection coating based on subwavelength surface Mie resonators,” Nat Commun **3**(692), 692 (2012). [CrossRef] [PubMed]

4. C. Lin and M. L. Povinelli, “Optical absorption enhancement in silicon nanowire arrays with a large lattice constant for photovoltaic applications,” Opt. Express **17**(22), 19371–19381 (2009). [CrossRef] [PubMed]

7. Z. Fan, R. Kapadia, P. W. Leu, X. Zhang, Y. L. Chueh, K. Takei, K. Yu, A. Jamshidi, A. A. Rathore, D. J. Ruebusch, M. Wu, and A. Javey, “Ordered arrays of dual-diameter nanopillars for maximized optical absorption,” Nano Lett. **10**(10), 3823–3827 (2010). [CrossRef] [PubMed]

8. L. Cao, J. S. White, J. S. Park, J. A. Schuller, B. M. Clemens, and M. L. Brongersma, “Engineering light absorption in semiconductor nanowire devices,” Nat. Mater. **8**(8), 643–647 (2009). [CrossRef] [PubMed]

10. K. Seo, M. Wober, P. Steinvurzel, E. Schonbrun, Y. Dan, T. Ellenbogen, and K. B. Crozier, “Multicolored Vertical Silicon Nanowires,” Nano Lett. **11**(4), 1851–1856 (2011). [CrossRef] [PubMed]

19. M. Heiss, E. Russo-Averchi, A. Dalmau-Mallorquí, G. Tütüncüoğlu, F. Matteini, D. Rüffer, S. Conesa-Boj, O. Demichel, E. Alarcon-Lladó, and A. Fontcuberta i Morral, “III-V nanowire arrays: growth and light interaction,” Nanotechnology **25**(1), 014015 (2014). [CrossRef] [PubMed]

21. G. Mariani, P. S. Wong, A. M. Katzenmeyer, F. Léonard, J. Shapiro, and D. L. Huffaker, “Patterned radial GaAs nanopillar solar cells,” Nano Lett. **11**(6), 2490–2494 (2011). [CrossRef] [PubMed]

22. T. Mårtensson, P. Carlberg, M. Borgstrom, L. Montelius, W. Seifert, and L. Samuelson, “Nanowire arrays defined by nanoimprint lithography,” Nano Lett. **4**(4), 699–702 (2004). [CrossRef]

22. T. Mårtensson, P. Carlberg, M. Borgstrom, L. Montelius, W. Seifert, and L. Samuelson, “Nanowire arrays defined by nanoimprint lithography,” Nano Lett. **4**(4), 699–702 (2004). [CrossRef]

4. C. Lin and M. L. Povinelli, “Optical absorption enhancement in silicon nanowire arrays with a large lattice constant for photovoltaic applications,” Opt. Express **17**(22), 19371–19381 (2009). [CrossRef] [PubMed]

26. B. C. P. Sturmberg, K. B. Dossou, L. C. Botten, A. A. Asatryan, C. G. Poulton, C. M. de Sterke, and R. C. McPhedran, “Modal analysis of enhanced absorption in silicon nanowire arrays,” Opt. Express **19**(S5Suppl 5), A1067–A1081 (2011). [CrossRef] [PubMed]

8. L. Cao, J. S. White, J. S. Park, J. A. Schuller, B. M. Clemens, and M. L. Brongersma, “Engineering light absorption in semiconductor nanowire devices,” Nat. Mater. **8**(8), 643–647 (2009). [CrossRef] [PubMed]

13. 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 Lett. **10**(2), 439–445 (2010). [CrossRef] [PubMed]

15. J. Kupec and B. Witzigmann, “Dispersion, wav propagation and efficiency analysis of nanowire solar cells,” Opt. Exp. **17**(12), 10399–10410 (2009). [CrossRef]

27. S. Hu, C. Chi, K. T. Fountaine, M. Yao, H. A. Atwater, P. D. Dapkus, N. S. Lewis, and C. Zhou, “Optical, electrical, and solar energy-conversion properties of gallium arsenide nanowire-array photoanodes,” Energy Environ. Sci. **6**(6), 1879–1890 (2013). [CrossRef]

29. Y. Yu and L. Cao, “Coupled leaky mode theory for light absorption in 2D, 1D, and 0D semiconductor nanostructures,” Opt. Express **20**(13), 13847–13856 (2012). [CrossRef] [PubMed]

30. Y. M. Chang, J. Shieh, and J. Y. Juang, “Subwavelength antireflective Si nanostructures fabricated by using the self-assembled silver metal-nanomask,” J. Phys. Chem. C **115**(18), 8983–8987 (2011). [CrossRef]

31. S. Patchett, M. Khorasaninejad, O. Nixon, and S. S. Saini, “Effective index approximation for ordered silicon nanowire arrays,” JOSA B **30**(2), 306–313 (2013). [CrossRef]

10. K. Seo, M. Wober, P. Steinvurzel, E. Schonbrun, Y. Dan, T. Ellenbogen, and K. B. Crozier, “Multicolored Vertical Silicon Nanowires,” Nano Lett. **11**(4), 1851–1856 (2011). [CrossRef] [PubMed]

32. E. Yablonovitch, “Statistical ray optics,” J. Opt. Soc. Am. **72**(7), 899–907 (1982). [CrossRef]

34. D. M. Callahan, J. N. Munday, and H. A. Atwater, “Solar cell light trapping beyond the ray optic limit,” Nano Lett. **12**(1), 214–218 (2012). [CrossRef] [PubMed]

7. Z. Fan, R. Kapadia, P. W. Leu, X. Zhang, Y. L. Chueh, K. Takei, K. Yu, A. Jamshidi, A. A. Rathore, D. J. Ruebusch, M. Wu, and A. Javey, “Ordered arrays of dual-diameter nanopillars for maximized optical absorption,” Nano Lett. **10**(10), 3823–3827 (2010). [CrossRef] [PubMed]

19. M. Heiss, E. Russo-Averchi, A. Dalmau-Mallorquí, G. Tütüncüoğlu, F. Matteini, D. Rüffer, S. Conesa-Boj, O. Demichel, E. Alarcon-Lladó, and A. Fontcuberta i Morral, “III-V nanowire arrays: growth and light interaction,” Nanotechnology **25**(1), 014015 (2014). [CrossRef] [PubMed]

35. S. L. Diedenhofen, O. T. A. Janssen, G. Grzela, E. P. A. M. Bakkers, and J. Gómez Rivas, “Strong geometrical dependence of the absorption of light in arrays of semiconductor nanowires,” ACS Nano **5**(3), 2316–2323 (2011). [CrossRef] [PubMed]

43. J. Y. Jung, Z. Guo, S. W. Jee, H. D. Um, K. T. Park, and J. H. Lee, “A strong antireflective solar cell prepared by tapering silicon nanowires,” Opt. Express **18**(S3Suppl 3), A286–A292 (2010). [CrossRef] [PubMed]

43. J. Y. Jung, Z. Guo, S. W. Jee, H. D. Um, K. T. Park, and J. H. Lee, “A strong antireflective solar cell prepared by tapering silicon nanowires,” Opt. Express **18**(S3Suppl 3), A286–A292 (2010). [CrossRef] [PubMed]

## 2. Results and discussion

### 2.1 Uniform nanowire arrays

27. S. Hu, C. Chi, K. T. Fountaine, M. Yao, H. A. Atwater, P. D. Dapkus, N. S. Lewis, and C. Zhou, “Optical, electrical, and solar energy-conversion properties of gallium arsenide nanowire-array photoanodes,” Energy Environ. Sci. **6**(6), 1879–1890 (2013). [CrossRef]

^{2}, far exceeding the absorbed current of 10.5 mA/cm

^{2}in a planar equivalent material volume (thickness = 150 nm). This strong absorption, despite the low fill fraction, is explained by coupling into resonant ‘leaky’ radial waveguide modes, which are electromagnetic modes with enhanced electric and magnetic field intensities localized on the nanowire. The spectral positions of the modal resonances depend only on radius and are independent of array period, leading to the denotation of the modes as radial modes. The eigenvalue equation for an individual, infinitely long cylinder shown below [Eq. (1)], as determined from Maxwell’s equations for a lossless, dielectric medium, determines the resonant wavelengths for each leaky mode, where

*a*is the wire radius,

*k*and

_{0}*β*are the total and

*z*-component of the wave vector in free space,

*k*(

_{1}*k*) and

_{2}*n*(

_{1}*n*) are the transverse components of the wave vectors and refractive indices inside(outside) the nanowire, and

_{2}*J*,

_{m}*K*, and

_{m}*H*are the Bessel (1st kind), modified Bessel (2nd kind), and Hankel (1st kind) functions. Leaky mode resonances occur at

_{m}*β*= 0, leading the first term on the right hand side of the equation to give the transverse magnetic (TM) modes and the second term to give the transverse electric (TE) modes.

_{mn}or TE

_{mn}, where

*m*and

*n*are the azimuthal and radial mode numbers, indicating the number of field maxima in a given direction. Specifically, in the previous work, the TM

_{11}and TM

_{12}modes, as determined from Eq. (1), were identified as the modes responsible for the resonant absorption peaks of nanowire arrays. In this array geometry (smaller radius), the TM

_{11}mode is observed at 675nm, matching the wavelength predicted by the eigenvalue equation. The absorbed power profile of the TM

_{11}resonance at 675 nm for the uniform nanowire is displayed as an inset in Fig. 1, matching the analytically-predicted profile. The TM

_{12}mode is no longer accessible within the wavelength range of interest for a nanowire with a radius of 65 nm due to the dispersion curve of GaAs; the increase in absorption in the blue end of the spectrum is simply attributable to the strong absorption of GaAs.

^{2}in the uniform nanowire array is impressive given the material volume, but needs to increase further to be competitive with the current world record thin film GaAs short circuit current of 29.7 mA/cm

^{2}or approach the 4n

^{2}Lambertian limit of 32.6 mA/cm

^{2}for 150 nm planar equivalence [44

44. M. A. Green, K. Emery, Y. Hishikawa, W. Warta, and E. D. Dunlop, “Solar cell efficiency tables (version 42),” Prog. Photovolt. Res. Appl. **21**(1), 827–837 (2013). [CrossRef]

### 2.2 Multi-radii nanowire arrays

_{11}modes of the nanowires, which red shift with increasing radius, as predicted by the eigenvalue equation [Eq. (1)]. The predicted resonant wavelength for the TM

_{11}modes of the 45, 55, 65, and 75 nm radii nanowires are 515, 590, 675, and 760 nm, respectively, as indicated in Fig. 2(b) and Fig. 2(c). In this case, the peaks in the absorption curve are slightly red-shifted with respect to the predicted resonant wavelengths. We expect that this is due to spectral overlap of the neighboring TM

_{11}resonances. Additionally, the absorption curves for the largest radii nanowire (r = 75 nm) exhibits a second absorption peak in the blue region, which corresponds to the TM

_{12}resonance, predicted to occur at 440 nm. Field profiles of nanowire cross sections within this array confirm coupling into the TM

_{11}and TM

_{12}modes, as discussed above. Therefore, we conclude that a multi-radii nanowire array increases the number of spectral resonances and results in broader absorption than for a uniform nanowire array.

### 2.3 Nanocone arrays

*z*, is equivalent to a varying radius coordinate, where

*z*= 3 µm corresponds to a radius of 40 nm and

*z*= 0 µm corresponds to a radius of 100nm. The largely red diagonal peak stretching from

*z*= 0 µm and

*λ*~900 nm up to

*z*= 3 µm and

*λ*~500 nm, the most prominent feature of Fig. 3(b), is the absorption into the resonant TM

_{11}mode. To confirm coupling into the TM

_{11}mode of the nanocone, radial cross sections of the absorbed power at the resonant wavelength-radius pairs indicated by the prominent diagonal peak in Fig. 3(b) were observed and found to match that of TM

_{11}modes. Additionally, these wavelength-radius pairs match the eigenvalues predicted by Eq. (1). In addition to the strong TM

_{11}peak observed in Fig. 3(b), a second, fainter diagonal peak is visible stretching from

*z*= 0 µm and

*λ*~500 nm up to

*z*~1.5 µm and

*λ*~450 nm. The TM

_{12}mode is responsible for this resonant absorption. The peak slowly fades away for larger

*z*as the nanocone radius decreases and ultimately disappears around

*r*= 70 nm (or

*z*= 1.5 µm) where the mode is no longer accessible due to the dispersion curve of GaAs. The diagonal character of both the TM

_{11}and TM

_{12}peaks demonstrates that these modes have a spectrum of resonant wavelengths in a nanocone, as intended.

*z*axis. This modulation in absorption is due to longitudinal resonances, and the overall absorbed power intensity profile of Fig. 3(b) is explained by a linear combination of longitudinal resonances and radial resonances. This phenomenon is more clearly discernible from

*xz*cross sections of the power absorbed in a nanocone, displayed in Fig. 3(c) for wavelengths of 400, 500, 600, 700, and 800 nm. All five of these cross sections illustrate the longitudinal modes present in the nanocone which give rise to the characteristic vertical oscillations in absorption intensity. Focusing on the four longer wavelength cross sections, the radial TM

_{11}resonance shifts downward to larger radius with increasing wavelength and has multiple lobes in the vertical direction due to its convolution with the longitudinal resonances. No strong radial mode is visible for the 400 nm wavelength cross section because GaAs absorbs strongly in this region and light does not penetrate deep enough into the nanocone to establish a radial mode. Additionally, in the 500 nm wavelength cross section, the character of the TM

_{12}mode is visible at the bottom of the nanocone, as an additional radial absorption peak becomes visible at the rim of the nanocone. From this detailed analysis of nanocone absorption, we conclude that arrays of truncated nanocones exhibit spectrally-extended resonances and provide another method to achieving a more broadband optical absorption response in nanowire arrays.

### 2.4 Optimization

*λ*= 675 nm) and their absorption curves are shown in Fig. 4(a), Fig. 4(b), and Fig. 4(c), respectively. The side-by-side display of the power absorption cross sections [Fig. 4(b)] in each optimized structure underscores their essentially identical TM

_{res}_{11}modal characteristic.

^{2}and the uniform array of 65 nm radius nanowires [Fig. 4(a)] absorbs 25.0 mA/cm

^{2}. The absorption curves for these cases are displayed in Fig. 1 and Fig. 4(c) as the black and red lines, respectively. Note that partial spectral averaging has been used for the planar layer to smooth out the Fabry-Perot resonances (see Methods for details). All nanostructures are positioned on top of an infinite Si substrate and embedded in a 30 nm layer of SiO

_{x}to emulate SAG-MOCVD as-grown structures [27

27. S. Hu, C. Chi, K. T. Fountaine, M. Yao, H. A. Atwater, P. D. Dapkus, N. S. Lewis, and C. Zhou, “Optical, electrical, and solar energy-conversion properties of gallium arsenide nanowire-array photoanodes,” Energy Environ. Sci. **6**(6), 1879–1890 (2013). [CrossRef]

_{11}and TM

_{12}resonances are distributed within the UV-Vis spectrum up to the bandgap of GaAs (λ ~350-900 nm). Using broadband simulations, the optimized case was determined to be a unit cell with wire radii of 50, 60, 70, and 80 nm; however, due to fitting errors in the imaginary part of the refractive index (see Methods for details), the final optimized multi-radii nanowire array was determined to be an array with radii of 45, 55, 65, and 75 nm, using single wavelength simulations. The absorption as a function of wavelength for this case is shown in Fig. 4(c) (blue line) and achieves an absorbed photocurrent of 28.8 mA/cm

^{2}, corresponding to a 15% improvement over the uniform nanowire array. As implied from the individual wire absorption curves in Fig. 2(c), the total array absorption curve has four strong peaks distributed relatively evenly across the visible spectrum and an extended shoulder around 450 nm, which correspond to absorption into the TM

_{11}modes corresponding to each wire radius and the TM

_{12}mode of the 75 nm radius wire, respectively. A previous study by Sturmberg et al. using a Bloch wave expansion to frame the modal analysis found that when multiple wire radii are included within a 30% fill fraction array of silicon nanowires, nearly a 30% increase in absorbance was found for a 2 x 2 unit cell array over a uniform array [41]. Additionally, they extended their study to include 4 x 4 unit cell arrays, and achieved an additional 4% improvement over the optimized 2 x 2 case. The findings of Sturmberg et al. indicate that larger unit cell arrays could yield additional improvement in absorbed current density. By contrast, our study focused on lower array fill fractions, such that the scattering cross sections of nearest neighbor wires overlap, but the scattering cross sections of second-nearest neighbors and beyond, do not overlap. In this sparse array limit, larger unit cells face diminishing returns because the period of the sub-lattice for each wire radii exceeds the wire scattering cross section, preventing incident light from coupling into the appropriate resonant structure.

^{2}and an 18% improvement over the uniform array absorption [Fig. 4(c), green line]. At these dimensions, the extended resonance spectrums of the two TM modes overlap by more than 50 nm, enabling the observed near-unity broadband absorption. The truncated nanocone absorption equals or exceeds that of the uniform array except in the region of the TM

_{11}resonance of the uniform array (650-750 nm). The higher absorption in the uniform array in this spectral region is due to a difference in vertical distance over which the mode is resonant: the entire length of the uniform wire compared to only a small fraction of the nanocone length.

## 3. Conclusion

45. P. Mohan, J. Motohisa, and T. Fukui, “Controlled growth of highly uniform, axial/radial direction-defined, individually addressable InP nanowire arrays,” Nanotech. **16**(12), 2903–2907 (2005). [CrossRef]

43. J. Y. Jung, Z. Guo, S. W. Jee, H. D. Um, K. T. Park, and J. H. Lee, “A strong antireflective solar cell prepared by tapering silicon nanowires,” Opt. Express **18**(S3Suppl 3), A286–A292 (2010). [CrossRef] [PubMed]

^{2}), both nanocone arrays and multi-radii wire arrays may be promising routes to improve the optoelectronic performance of semiconductor nanowire arrays as solar cells.

## 4. Methods

*x*and

*y*directions and infinite boundary conditions, rendered as perfectly matched layers (PML), in the

*z*direction. All nanowire structures were modeled as GaAs, using the Palik material data provided by Lumerical, and were anchored on an infinite silicon substrate with a thin, 30nm layer of silicon oxide to emulate as-grown wires from previous work [27

**6**(6), 1879–1890 (2013). [CrossRef]

^{2}, was calculated from the absorption as a function of wavelength by weighting the simulated absorption curve by the AM1.5G spectrum and integrating over wavelength. Plots of normalized power absorbed were calculated by recording the electric field intensity spatially and multiplying by the imaginary part of the permittivity of GaAs.

## Acknowledgments

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**OCIS Codes**

(160.6000) Materials : Semiconductor materials

(220.2740) Optical design and fabrication : Geometric optical design

(350.6050) Other areas of optics : Solar energy

(350.4238) Other areas of optics : Nanophotonics and photonic crystals

**ToC Category:**

Light Trapping for Photovoltaics

**History**

Original Manuscript: February 10, 2014

Revised Manuscript: April 6, 2014

Manuscript Accepted: April 6, 2014

Published: April 18, 2014

**Citation**

Katherine T. Fountaine, Christian G. Kendall, and Harry A. Atwater, "Near-unity broadband absorption designs for semiconducting nanowire arrays via localized radial mode excitation," Opt. Express **22**, A930-A940 (2014)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-S3-A930

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