An Efficient and Effective Design of InP Nanowires for Maximal Solar Energy Harvesting
Solar cells based on subwavelength-dimensions semiconductor nanowire (NW) arrays promise a comparable or better performance than their planar counterparts by taking the advantages of strong light coupling and light trapping. In this paper, we present an accurate and time-saving analytical design for optimal geometrical parameters of vertically aligned InP NWs for maximal solar energy absorption. Short-circuit current densities are calculated for each NW array with different geometrical dimensions under solar illumination. Optimal geometrical dimensions are quantitatively presented for single, double, and multiple diameters of the NW arrays arranged both squarely and hexagonal achieving the maximal short-circuit current density of 33.13 mA/cm2. At the same time, intensive finite-difference time-domain numerical simulations are performed to investigate the same NW arrays for the highest light absorption. Compared with time-consuming simulations and experimental results, the predicted maximal short-circuit current densities have tolerances of below 2.2% for all cases. These results unambiguously demonstrate that this analytical method provides a fast and accurate route to guide high performance InP NW-based solar cell design.
KeywordsComputational modeling III–V semiconductor materials Photovoltaic cells
Full width at half maximum
For future generation solar cells, semiconductor nanowire (NW) arrays have unraveled a new pathway to greatly reduce material consumption and fabrication cost while maintaining or even improving the device performance as compared with their thin film or bulk counterparts [1, 2]. This fascinating feature is largely attributed to the remarkable optical properties of the NWs, including increased absorption [3, 4] and spectral selectivity [5, 6, 7]. Among various III–V materials, InP NW arrays have attracted intensive research effort for solar cell application due to the direct bandgap and low intrinsic surface recombination velocity . Up to date, the highest energy conversion efficiency achieved 13.8% for InP NW arrays in a cell of 1 mm2 in area .
Since the optical properties of NW arrays can be distinctively adjusted by tuning their three-dimensional geometry, to further improve the performance of NW-based solar cells, great attention has been put on how to optimize the morphology and topology of III–V NW arrays to maximize the light absorption [5, 9, 10, 11, 12, 13]. Specifically, the NWs’ diameter, periodicity, and arrangement have been investigated to maximize the absorption of solar energy [6, 14, 15, 16]. It is reported that tuning the diameter of the NW will change the optical modes existed within the NW. This will lead to localized light absorption maxima for those incident wavelengths corresponding to the respective resonant modes [5, 6, 17, 18]. Also, NW arrays with optimized periodicity or filling ratio (FR) can suppress the reflection and transmission while enhancing the scattering to the incident light resulting in the prolonged optical path and thus the enhanced light absorption [19, 20, 21]. Besides, Martin Foldyna et al. have concluded that the dependence of the light absorption on the arrangement of the NW arrays is rather small since the light trapping effect of NWs is based on the individual waveguiding when the light coupling among neighboring NWs is neglected .
To find the maximal solar energy harvesting, the effect of the three-dimensional parameters and the arrangement of the NW arrays should be considered together. However, most of the reported optimal geometrical dimensions and arrangement of NW arrays for maximal solar spectrum harvesting are still parameter-space-determinated local optima. Besides, the incident solar spectrum combining with material dispersive properties add more difficulty to analytically solve this problem. Therefore, intensive and time-consuming numerical simulations such as finite-difference time-domain (FDTD) are frequently adopted to address this multi-parameters optimization problem. Sturmberg et al. reported a semi-analytic method to narrow down the range of the optimal dimensions of single diameter NW arrays . Although this method is applicable for various materials, FDTD simulations should still be accompanied to find the exact optimal values. Moreover, this method is less helpful for superb absorber combined with multi-radii NW arrays .
In this paper, we present an analytical design for optimal geometrical dimensions of single, double, and multiple diameter InP NW arrays to maximize solar energy absorption. Diameters of NWs are determined by leaky mode resonance and Mie theory whereas the periodicities are identified by construction of an effective medium layer to minimize light reflection and transmission. Squarely and hexagonal distributed NW arrays are both considered. Moreover, intensive FDTD simulations are accompanied to verify the effectiveness of our method. The well matching of the largest short-circuit current densities generated from the NW arrays with the calculated geometrical parameters and the values obtained from FDTD simulations prove the effectiveness of the proposed method to guide the practical NW-based photovoltaic cells design.
Design for Maximal Light Harvesting of InP NWs
In order to analytically determine each geometrical parameter of NW arrays, the multiple-parameter optimization problem for maximal light harvesting is decomposed into two processes: (1) NWs’ diameter-determinant resonant mode control and (2) FR-affected minimal reflectance and transmittance of incident solar energy. We construct the relationship of individual geometrical parameter with respective determinant process and identify each optimal value leading to maximal light absorption. Double diameter NW arrays are chosen as the design example for illustration of the proposed method. Optimal geometrical dimensions of single diameter NW arrays as a simpler case can also be acquired during the derivation. The diameter and periodicity for four diameter NW arrays can also be calculated as an extension of the example. For squarely arranged double diameters NW arrays, the diameters of the diagonal NWs have the same value as D major and the diameters of the rest two NWs are named as D supplementary. For hexagonal arranged NW arrays, the diameter of the center NW is D major and the diameters of the NWs at the peripheral are D supplementary.
It is reported that NW arrays can support leaky/guided resonance modes, each of which lead to strong absorption peaks. Besides, the fundamental nature of waveguide suggests that the mode number grows with the rise of the diameter of NW. Consequently, the optimal diameter of NW should be large enough to support more modes so as to include larger number of absorption resonances. However, too large diameters of NWs are less preferable since the higher order modes they supported possess more nodes which couple less efficiently to the incident plane waves . Besides, the material property and the incident solar spectrum place other limitations on the selection of the optimal diameter. Only when the resonant modes lie within the absorption region, they can contribute to the photocurrent. The absorption region is defined by the superposition of the material absorbing range of up to critical wavelength and the incident AM 1.5G spectrum .
As a result, to quantitatively determine the D major of the NW arrays, leaky mode resonance is initially adopted to calculate the respective resonant wavelengths for different diameters of NWs . This gives the distribution of the resonant modes in the absorption region. Therefore, the optimal D major should support two modes to satisfy all of the above criterions. Secondly, Mie theory is adopted to calculate the normalized absorption efficiencies of those NWs in step one. Strictly speaking, Mie theory cannot be applied to the situation when the incident wave vector aligned perfectly parallel to the axis of NWs since the eigenvalue equation is ill-defined . However, this situation can be approximated as the glazing incident of incoming light (very small incident angle θ with respect to the axis of NW) since at the interface of NW arrays, the wave front of the incident light will be perturbed by the high index of NWs which introduces transverse components to the wave vector allowing the adoption of Mie theory . Therefore, the optimal D major are the one who support two modes while keeping the full width at half maximum (FWHM) of the lowest resonant mode in the normalized absorption efficiency spectrum within the absorption region. After acquisition of D major, the D supplementary is calculated on the condition that the NWs should support one mode for reducing reflection and material saving and their resonant wavelength should match the valley of the D major’s normalized absorption efficiency spectrum.
The periodicity of the NW arrays can be computed by construction of an effective medium layer. This artificial layer represents the reflection and transmission behavior of the NW arrays which is only related to the material FR. As a result, the diameter, periodicity, and the arrangement of the NW arrays are removed from the calculation. In this way, the transmittance and reflectance of NW arrays can be evaluated by applying Fresnel equations on this effective medium layer and therefore the optimal FR can be analyzed. Based on the relationship of FR and periodicity, the periodicities for both hexagonal and squarely arrangement NW arrays are obtained. Detailed description of our proposed method is presented in the following sections.
A. Optimal Diameters of InP NW Arrays for Maximal Light Harvesting
To increase the light absorption, the number of resonant modes leading to strong absorption peaks should be maximized within the absorption region. On the blue end of the absorption region, incident AM 1.5G spectrum confines 300 nm as the high energy region. The critical wavelength λ c of 925 nm (bandgap of InP 1.34 eV) limits the red end of the absorbing region. As a result, it is proved that the InP NWs that support two resonant modes locating inside the absorbing region are able to best improve the light absorption . We expand this conclusion and use Mie theory to calculate the exact value.
After acquisition the Q abs of the HE11 mode, the FWHM of respective diameter of NWs can be found out, and therefore, the optimal diameter for maximal light harvesting is determined. Upon decision of the major diameter, the supplementary diameter is confirmed on the condition that its normalized absorption peak wavelength should match the normalized absorption efficiency valley of the major diameter. For four diameter NW arrays, the third and fourth diameters are determined in a similar way. Their normalized absorption efficiency peaks should match the valleys of the superposition of normalized absorption efficiency spectrum of the primary and secondary NWs. It is noteworthy that except for the major NWs, the second, third, and fourth NWs are desired to support only one mode since the small diameter size can both reduce the reflectance at the air-NW interface and reduce material consumption.
B. Optimal FR of InP NW Arrays for Maximal Light Harvesting
Results and Discussion
Single and multiple diameters of InP NW arrays of squarely and hexagonal arrangements demonstrate the validity of the proposed method. Meanwhile, FDTD numerical simulations (Lumerical FDTD Solutions 8.15) are also provided to compare with our method. Periodic boundary condition is applied along x and y axes while perfect matching condition is set along z axis as illustrated in Fig. 1. The InP NWs are vertically standing on SiO2 substrate. The optical constants for InP and SiO2 are from Palik material data provided by Lumerical. The parameter space for diameters of NWs ranges from 50 to 200 nm whereas the FR is from 0.05 to the possible maximal values for squarely and hexagonal NWs.
A. Maximal Light Harvesting for Single Diameter InP NWs
B. Maximal Light Harvesting for Double Diameter InP NWs
C. Maximal Light Harvesting for Four Diameter InP NWs
In this study, we present model for effective and fast design of both squarely and hexagonal InP NW arrays to achieve the highest light harvesting for photovoltaic application. Geometrical dimensions for vertically aligned single, double, and multiple diameters of NW arrays are investigated. Compared with time-consuming FDTD simulations, our predicted maximal short-circuit current densities with calculated three-dimensional NW arrays remain tolerances below 2.2% for all cases. For single diameter NW arrays, the optimal diameter is 184 nm which is only 4 nm difference to the reported highest efficiency InP NW solar cells. In the multiple diameter NW arrays, the diameters of the rest of NWs are optimized to satisfy the conditions that they support only one resonant mode and the corresponding wavelengths match the absorption valley of the major NWs. Moreover, the FR of the NW array is optimized to be 0.2 by creating an effective medium layer which is regardless of the diameter, periodicity, and arrangements of NWs. Compared with the optical modeling, the predicted highest short-circuit current densities for single diameter NW arrays lie within 0.33 and 0.1% tolerance for squarely and hexagonal NW array. The arrangements of NW array have little influence on the light absorption with optimal geometrical parameters, but the coupling among neighboring NWs becomes serious for multiple diameter NWs at large FR value. Squarely arranged four diameter NW arrays were also presented and the highest short-circuit current densities predicted to be 33.13 mA/cm2 with a low tolerance of 2.2%. The time-efficient, high precision with wide suitability of the proposed design for InP NW arrays demonstrate itself to be a promising tool to guide practical NW-based solar cell design.
This work was supported by the Academic Research Fund (RG97/14) of the Ministry of Education of Singapore; National Natural Science Foundation of China (51402148 and 11304147); National Key Research Project administrated by the Ministry of Science and Technology of China (2016YFB0401702), Guangdong High Tech Project (2014A010105005 and 2014TQ01C494); Shenzhen Innovation Project (KC2014JSQN0011A, JCYJ20150630145302223, JCYJ20150529152146471, and JCYJ20160301113537474) and Foshan Innovation Project (2014IT100072); and Shenzhen Key Laboratory Project (ZDSYS201602261933302).
DW drafted the manuscript and developed the algorithm. XHT supervised and coordinated the projects. KW and ZBH revised the manuscript. XQL helped with FDTD simulation. All authors read and approved the final manuscript.
The authors declare that they have no competing interests.
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
- 9.Wallentin J, Anttu N, Asoli D, Huffman M, Åberg I, Magnusson MH, Siefer G, Fuss-Kailuweit P, Dimroth F, Witzigmann B, Xu HQ, Samuelson L, Deppert K, Borgström MT (2013) InP nanowire array solar cells achieving 13.8% efficiency by exceeding the ray optics limit. Science 339:1057–1060CrossRefGoogle Scholar
- 17.Paniagua-Dominguez R, Grzela G, Rivas JG, Sanchez-Gil JA (2013) Enhanced and directional emission of semiconductor nanowires tailored through leaky/guided modes. Nano 5:10582–10590Google Scholar
- 20.Li J, Yu H, Wong SM, Li X, Zhang G, Lo PG-Q, Kwong D-L (2009) Design guidelines of periodic Si nanowire arrays for solar cell application. Adv Mater 95:243113Google Scholar
- 27.Standard tables for reference solar spectral irradiance at air mass 1.5: direct normal and hemispherical for a 37 degree tilted surface, ISO 9845-1. http://rredc.nrel.gov/solar/spectra/am1.5/.
- 30.Bohren CF, Huffman DR (1983) Absorption and scattering of light by small particles. Wiley, New YorkGoogle Scholar
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.