# Absorption of Light in Finite Semiconductor Nanowire Arrays and the Effect of Missing Nanowires

^{1}

^{2}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

^{2}NWs in total, which we term an N × N array (see Figure 1a for a schematic). We wish to investigate the absorption in the center NW and hence limit the study to an odd N. When studying the effect of a missing NW in the array, we consider a supercell consisting of 14 × 14 NWs, which is periodically repeated. In this supercell, the center NW is missing (see Figure 1b for a schematic).

_{2v}symmetry is a highly symmetric problem that can be easily symmetry reduced [14]. We can speed up the simulations by using the x-z and y-z mirror planes through the center NW to reduce the simulation domain to ¼ of the original size [14]. We model the results for x-polarized incident light. The x-z mirror plane is then set to a perfect magnetic conductor (PMC) and the y-z mirror plane to a perfect electric conductor (PEC) [14]. For the finite array (Figure 1a), we terminate the simulation domain in the x-y plane outside of the NW array with perfectly matched layers (PMLs). For the supercell with the missing NW (Figure 1b), we use PEC and PMC boundary conditions at the edges of the simulation domain in the x-y plane to give, for the normally incident light, a periodic repetition of the supercell [14].

#### 2.1. FEM Simulation Settings

#### 2.2. Absorption Cross-Section and Short-Circuit Current

^{2}direct and circumsolar spectrum [16]. For the lower limit, ${\lambda}_{\mathrm{low}}$, in the integration, we use 300 nm, below which the incident photon flux is negligible (Figure 2a). For the upper limit, we use the bandgap wavelength ${\lambda}_{\mathrm{bg}}=925$ nm, assuming a bandgap energy of 1.34 eV for the InP [17]. Note that in Equation (1), we thus assume that each absorbed above-bandgap photon contributes one electric charge carrier to the short-circuit current. Note that we focus here on absorption in the NWs without taking into account a possible contribution to the photocurrent from absorption in the substrate. Such an approach is motivated by the typically small contribution to the photocurrent from absorption in the substrate in NW array solar cells with an optimized diode configuration [18,19].

## 3. Results

^{2}, close to the maximum value of 31.1 mA/cm

^{2}for InP, which is obtained when ${\sigma}_{\mathrm{abs}}\left(\lambda \right)={P}^{2}$ for $300<\lambda <925$ nm.

_{11}waveguide mode in the NW, as detailed in [15]. The peak drops to ${\sigma}_{\mathrm{abs}}=2.6{P}^{2}$ at N = 3 and, at N = 5, the highest ${\sigma}_{\mathrm{abs}}$ is equal to 1.1${P}^{2}$ at $\lambda =710$ nm. As seen from Figure 2b, the single NW absorbs light much stronger than the NW in the infinite array, but already, at N = 5, the center NW shows absorption characteristics close to those of the NWs in the infinite array. This fast convergence of the absorption in the center NW with an increasing N is seen also in the ${I}_{\mathrm{sc}}$ (red squares in Figure 3). At N = 1, the center NW shows ${I}_{\mathrm{sc}}=116$ pA, which decreases to 55.9 pA at N = 3. At N = 5, ${I}_{\mathrm{sc}}=$ 47.3 pA for the center NW, which is just 3% (relative) higher than the 45.8 pA of the NWs in the infinite array.

_{sc}in each NW). The increase in the I

_{sc}in all the neighboring NWs is 35.2 pA compared to the case of the array without the missing NW. The increased absorption occurs predominantly in the four closest neighbors to the missing NW with an increase in I

_{sc}by 6.9 pA in each of them and hence these four NWs contribute 27.6 pA of the increase, which is 78% of the total increase of 35.2 pA. The increase in I

_{sc}is virtually negligible when moving more than two NWs away from the missing NW (see Table A4: this fast convergence toward the values of the infinite array shows that our choice to use the supercell with 14 × 14 NWs is large enough to avoid the effects from a finite-sized supercell). Thus, we see a redistribution of absorption to neighboring NWs, an effect that is seen also when designing aperiodic NW arrays for absorption [20,21].

_{sc}due to the missing NW is 45.8–35.2 = 10.6 pA, which is 23% of the I

_{sc}of 45.8 pA that would be generated in the NW if it was not missing. In other words, the neighboring NWs manage to compensate for 77% of the expected I

_{sc}drop. On the other hand, if the missing NW is still present but electrical contacting to it failed, the NW would absorb sunlight corresponding to 45.8 pA but the NW would not be able to generate any short-circuit current from it. Hence, we conclude that a non-contacted NW is, from the optics point of view, a four times more severe defect for I

_{sc}than a missing NW.

_{sc}is then given by $4M{I}_{\mathrm{sc},\mathrm{compensate}}$ and the fraction of compensation is given by $4M{I}_{\mathrm{sc},\mathrm{compensate}}/\left({M}^{2}{I}_{\mathrm{sc},\mathrm{inf}}\right)$ with ${I}_{\mathrm{sc},\mathrm{inf}}=45.8$ pA, the above-stated current in each NW of the infinite array without defects. From Figure 5, we see that this estimate works very well already for M ≥ 3 and thus explains quantitatively the drop in the compensation with an increasing M.

## 4. Discussion

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Table A1.**I

_{sc}in pA for each NW in a finite 3 × 3 NW array; that is, N = 3. The column and row index of a cell in this table correspond with the position of the respective NW in the array. For convenience, the center NW is marked in bold and underlined.

71.7 | 56.6 | 71.7 |

56.6 | 55.9 | 56.6 |

71.7 | 56.6 | 71.7 |

**Table A2.**I

_{sc}in pA for each NW in a finite 5 × 5 NW array; that is, N = 5. The column and row index of a cell in this table correspond with the position of the respective NW in the array. For convenience, the center NW is marked in bold and underlined.

71.1 | 55.6 | 55.6 | 55.6 | 71.1 |

55.6 | 49.4 | 47.4 | 49.4 | 55.6 |

55.6 | 47.4 | 47.3 | 47.4 | 55.6 |

55.6 | 49.4 | 47.4 | 49.4 | 55.6 |

71.1 | 55.6 | 55.6 | 55.6 | 71.1 |

**Table A3.**I

_{sc}in pA for each NW in a finite 15 × 15 NW array; that is, N = 15. The column and row index of a cell in this table correspond with the position of the respective NW in the array. Note the eight-fold symmetry due to the mirroring around the horizontal line, the vertical line and both diagonals that go through the center NW. For convenience, the center NW is marked in bold and underlined. The grayed-out cells mark the NWs that are used for the calculation of I

_{sc,compensate}in Figure 5.

71.0 | 55.5 | 55.3 | 55.0 | 55.0 | 55.0 | 55.0 | 55.0 | 55.0 | 55.0 | 55.0 | 55.0 | 55.3 | 55.5 | 71.0 |

55.5 | 48.9 | 46.9 | 47.2 | 47.0 | 47.0 | 47.0 | 47.0 | 47.0 | 47.0 | 47.0 | 47.2 | 46.9 | 48.9 | 55.5 |

55.3 | 46.9 | 46.2 | 46.2 | 46.0 | 46.1 | 46.0 | 46.0 | 46.0 | 46.1 | 46.0 | 46.2 | 46.2 | 46.9 | 55.3 |

55.0 | 47.2 | 46.2 | 46.2 | 46.0 | 46.0 | 46.0 | 46.0 | 46.0 | 46.0 | 46.0 | 46.2 | 46.2 | 47.2 | 55.0 |

55.0 | 47.0 | 46.0 | 46.0 | 45.8 | 45.9 | 45.8 | 45.9 | 45.8 | 45.9 | 45.8 | 46.0 | 46.0 | 47.0 | 55.0 |

55.0 | 47.0 | 46.1 | 46.0 | 45.9 | 45.9 | 45.9 | 45.9 | 45.9 | 45.9 | 45.9 | 46.0 | 46.1 | 47.0 | 55.0 |

55.0 | 47.0 | 46.0 | 46.0 | 45.8 | 45.9 | 45.8 | 45.9 | 45.8 | 45.9 | 45.8 | 46.0 | 46.0 | 47.0 | 55.0 |

55.0 | 47.0 | 46.0 | 46.0 | 45.9 | 45.9 | 45.9 | 45.9 | 45.9 | 45.9 | 45.9 | 46.0 | 46.0 | 47.0 | 55.0 |

55.0 | 47.0 | 46.0 | 46.0 | 45.8 | 45.9 | 45.8 | 45.9 | 45.8 | 45.9 | 45.8 | 46.0 | 46.0 | 47.0 | 55.0 |

55.0 | 47.0 | 46.1 | 46.0 | 45.9 | 45.9 | 45.9 | 45.9 | 45.9 | 45.9 | 45.9 | 46.0 | 46.1 | 47.0 | 55.0 |

55.0 | 47.0 | 46.0 | 46.0 | 45.8 | 45.9 | 45.8 | 45.9 | 45.8 | 45.9 | 45.8 | 46.0 | 46.0 | 47.0 | 55.0 |

55.0 | 47.2 | 46.2 | 46.2 | 46.0 | 46.0 | 46.0 | 46.0 | 46.0 | 46.0 | 46.0 | 46.2 | 46.2 | 47.2 | 55.0 |

55.3 | 46.9 | 46.2 | 46.2 | 46.0 | 46.1 | 46.0 | 46.0 | 46.0 | 46.1 | 46.0 | 46.2 | 46.2 | 46.9 | 55.3 |

55.5 | 48.9 | 46.9 | 47.2 | 47.0 | 47.0 | 47.0 | 47.0 | 47.0 | 47.0 | 47.0 | 47.2 | 46.9 | 48.9 | 55.5 |

71.0 | 55.5 | 55.3 | 55.0 | 55.0 | 55.0 | 55.0 | 55.0 | 55.0 | 55.0 | 55.0 | 55.0 | 55.3 | 55.5 | 71.0 |

**Table A4.**I

_{sc}in pA for each NW in a 14 × 14 NW supercell; that is, N = 14. The center nanowire is missing, see Figure 1b for a schematic (we show here a repeat of the nanowires at the edge of the supercell in both the x and the y-direction, giving 15 × 15 values for the table). The column and row index of a cell in this table correspond with the position of the respective NW in the array. For convenience, the missing center NW is grayed out. Note the eight-fold symmetry due to the mirroring around the horizontal line, the vertical line and both diagonals that go through the missing center NW.

45.8 | 45.8 | 45.8 | 45.8 | 45.8 | 45.8 | 45.8 | 45.8 | 45.8 | 45.8 | 45.8 | 45.8 | 45.8 | 45.8 | 45.8 |

45.8 | 45.8 | 45.8 | 45.8 | 45.8 | 45.8 | 45.8 | 45.8 | 45.8 | 45.8 | 45.8 | 45.8 | 45.8 | 45.8 | 45.8 |

45.8 | 45.8 | 45.8 | 45.8 | 45.8 | 45.8 | 45.8 | 45.8 | 45.8 | 45.8 | 45.8 | 45.8 | 45.8 | 45.8 | 45.8 |

45.8 | 45.8 | 45.8 | 45.8 | 45.8 | 45.8 | 45.8 | 45.9 | 45.8 | 45.8 | 45.8 | 45.8 | 45.8 | 45.8 | 45.8 |

45.8 | 45.8 | 45.8 | 45.8 | 45.8 | 45.8 | 45.9 | 45.9 | 45.9 | 45.8 | 45.8 | 45.8 | 45.8 | 45.8 | 45.8 |

45.8 | 45.8 | 45.8 | 45.8 | 45.8 | 45.9 | 46.0 | 46.5 | 46.0 | 45.9 | 45.8 | 45.8 | 45.8 | 45.8 | 45.8 |

45.8 | 45.8 | 45.8 | 45.8 | 45.9 | 46.0 | 46.9 | 52.7 | 46.9 | 46.0 | 45.9 | 45.8 | 45.8 | 45.8 | 45.8 |

45.8 | 45.8 | 45.8 | 45.9 | 45.9 | 46.5 | 52.7 | 0 | 52.7 | 46.5 | 45.9 | 45.9 | 45.8 | 45.8 | 45.8 |

45.8 | 45.8 | 45.8 | 45.8 | 45.9 | 46.0 | 46.9 | 52.7 | 46.9 | 46.0 | 45.9 | 45.8 | 45.8 | 45.8 | 45.8 |

45.8 | 45.8 | 45.8 | 45.8 | 45.8 | 45.9 | 46.0 | 46.5 | 46.0 | 45.9 | 45.8 | 45.8 | 45.8 | 45.8 | 45.8 |

45.8 | 45.8 | 45.8 | 45.8 | 45.8 | 45.8 | 45.9 | 45.9 | 45.9 | 45.8 | 45.8 | 45.8 | 45.8 | 45.8 | 45.8 |

45.8 | 45.8 | 45.8 | 45.8 | 45.8 | 45.8 | 45.8 | 45.9 | 45.8 | 45.8 | 45.8 | 45.8 | 45.8 | 45.8 | 45.8 |

45.8 | 45.8 | 45.8 | 45.8 | 45.8 | 45.8 | 45.8 | 45.8 | 45.8 | 45.8 | 45.8 | 45.8 | 45.8 | 45.8 | 45.8 |

45.8 | 45.8 | 45.8 | 45.8 | 45.8 | 45.8 | 45.8 | 45.8 | 45.8 | 45.8 | 45.8 | 45.8 | 45.8 | 45.8 | 45.8 |

45.8 | 45.8 | 45.8 | 45.8 | 45.8 | 45.8 | 45.8 | 45.8 | 45.8 | 45.8 | 45.8 | 45.8 | 45.8 | 45.8 | 45.8 |

## References

- Czaban, J.A.; Thompson, D.A.; LaPierre, R.R. GaAs Core−Shell Nanowires for Photovoltaic Applications. Nano Lett.
**2009**, 9, 148–154. [Google Scholar] [CrossRef] - Wallentin, J.; Anttu, N.; Asoli, D.; Huffman, M.; Åberg, I.; Magnusson, M.H.; Siefer, G.; Fuss-Kailuweit, P.; Dimroth, F.; Witzigmann, B.; et al. InP Nanowire Array Solar Cells Achieving 13.8% Efficiency by Exceeding the Ray Optics Limit. Science
**2013**, 339, 1057–1060. [Google Scholar] [CrossRef][Green Version] - Cui, Y.; Wang, J.; Plissard, S.R.; Cavalli, A.; Vu, T.T.T.; van Veldhoven, R.P.J.; Gao, L.; Trainor, M.; Verheijen, M.A.; Haverkort, J.E.M.; et al. Efficiency Enhancement of InP Nanowire Solar Cells by Surface Cleaning. Nano Lett.
**2013**, 13, 4113–4117. [Google Scholar] [CrossRef] - Yao, M.; Huang, N.; Cong, S.; Chi, C.-Y.; Seyedi, M.A.; Lin, Y.-T.; Cao, Y.; Povinelli, M.L.; Dapkus, P.D.; Zhou, C. GaAs Nanowire Array Solar Cells with Axial p–i–n Junctions. Nano Lett.
**2014**, 14, 3293–3303. [Google Scholar] [CrossRef] - LaPierre, R.R.; Robson, M.; Azizur-Rahman, K.M.; Kuyanov, P. A Review of III–V Nanowire Infrared Photodetectors and Sensors. J. Phys. D Appl. Phys.
**2017**, 50, 123001. [Google Scholar] [CrossRef] - Kupec, J.; Witzigmann, B. Dispersion, Wave Propagation and Efficiency Analysis of Nanowire Solar Cells. Opt. Express
**2009**, 17, 10399–10410. [Google Scholar] [CrossRef] [PubMed] - Diedenhofen, S.L.; Janssen, O.T.A.; Grzela, G.; Bakkers, E.P.A.M.; Gómez Rivas, J. Strong Geometrical Dependence of the Absorption of Light in Arrays of Semiconductor Nanowires. ACS Nano
**2011**, 5, 2316–2323. [Google Scholar] [CrossRef] - Huang, N.; Lin, C.; Povinelli, M.L. Broadband Absorption of Semiconductor Nanowire Arrays for Photovoltaic Applications. J. Opt.
**2012**, 14, 024004. [Google Scholar] [CrossRef][Green Version] - Anttu, N.; Xu, H.Q. Efficient Light Management in Vertical Nanowire Arrays for Photovoltaics. Opt. Express
**2013**, 21, A558–A575. [Google Scholar] [CrossRef] - Otnes, G.; Borgström, M.T. Towards High Efficiency Nanowire Solar Cells. Nano Today
**2017**, 12, 31–45. [Google Scholar] [CrossRef][Green Version] - Barrigón, E.; Heurlin, M.; Bi, Z.; Monemar, B.; Samuelson, L. Synthesis and Applications of III–V Nanowires. Chem. Rev.
**2019**, 119, 9170–9220. [Google Scholar] [CrossRef] - Barrigón, E.; Zhang, Y.; Hrachowina, L.; Otnes, G.; Borgström, M.T. Unravelling Processing Issues of Nanowire-Based Solar Cell Arrays by Use of Electron Beam Induced Current Measurements. Nano Energy
**2020**, 71, 104575. [Google Scholar] [CrossRef] - Glembocki, O.J.; Piller, H. Indium Phosphide (InP). In Handbook of Optical Constants of Solids; Palik, E.D., Ed.; Academic Press: Burlington, VT, USA, 1997; pp. 503–516. ISBN 978-0-12-544415-6. [Google Scholar]
- Mäntynen, H.; Lipsanen, H.; Anttu, N. Symmetry Reduction in FEM Optics Modeling of Single and Periodic Nanostructures. Symmetry
**2021**, 13, 752. [Google Scholar] [CrossRef] - Anttu, N. Absorption of Light in a Single Vertical Nanowire and a Nanowire Array. Nanotechnology
**2019**, 30, 104004. [Google Scholar] [CrossRef][Green Version] - ASTM G173-03(2020). Standard Tables for Reference Solar Spectral Irradiances: Direct Normal and Hemispherical on 37° Tilted Surface; ASTM International: West Conshohocken, PA, USA, 2020. [Google Scholar]
- Vurgaftman, I.; Meyer, J.R.; Ram-Mohan, L.R. Band Parameters for III–V Compound Semiconductors and Their Alloys. J. Appl. Phys.
**2001**, 89, 5815–5875. [Google Scholar] [CrossRef][Green Version] - Otnes, G.; Barrigón, E.; Sundvall, C.; Svensson, K.E.; Heurlin, M.; Siefer, G.; Samuelson, L.; Åberg, I.; Borgström, M.T. Understanding InP Nanowire Array Solar Cell Performance by Nanoprobe-Enabled Single Nanowire Measurements. Nano Lett.
**2018**, 18, 3038–3046. [Google Scholar] [CrossRef] [PubMed] - Jain, V.; Nowzari, A.; Wallentin, J.; Borgström, M.T.; Messing, M.E.; Asoli, D.; Graczyk, M.; Witzigmann, B.; Capasso, F.; Samuelson, L.; et al. Study of Photocurrent Generation in InP Nanowire-Based P+-i-N+ Photodetectors. Nano Res.
**2014**, 7, 544–552. [Google Scholar] [CrossRef] - Lin, C.; Povinelli, M.L. Optimal Design of Aperiodic, Vertical Silicon Nanowire Structures for Photovoltaics. Opt. Express
**2011**, 19, A1148–A1154. [Google Scholar] [CrossRef][Green Version] - Sturmberg, B.C.P.; Dossou, K.B.; Botten, L.C.; Asatryan, A.A.; Poulton, C.G.; McPhedran, R.C.; Sterke, C.M. de Absorption Enhancing Proximity Effects in Aperiodic Nanowire Arrays. Opt. Express OE
**2013**, 21, A964–A969. [Google Scholar] [CrossRef] [PubMed][Green Version] - Rodriguez, S.R.K.; Schaafsma, M.C.; Berrier, A.; Gómez Rivas, J. Collective Resonances in Plasmonic Crystals: Size Matters. Phys. B Condens. Matter
**2012**, 407, 4081–4085. [Google Scholar] [CrossRef][Green Version] - Zundel, L.; Manjavacas, A. Finite-Size Effects on Periodic Arrays of Nanostructures. J. Phys. Photonics
**2018**, 1, 015004. [Google Scholar] [CrossRef][Green Version] - Dagytė, V.; Anttu, N. Modal Analysis of Resonant and Non-Resonant Optical Response in Semiconductor Nanowire Arrays. Nanotechnology
**2018**, 30, 025710. [Google Scholar] [CrossRef] [PubMed]

**Figure 1.**Schematics of (

**a**) a finite 15 × 15 NW array and (

**b**) 14 × 14 NWs in a supercell with the center NW missing. Note that the supercell in (

**b**) is repeated periodically in the x-y plane in the modelling.

**Figure 2.**(

**a**) Photon flux in the AM1.5D 900 W/m

^{2}spectrum. The inset shows the incident photon flux in an extended wavelength range. The absorption cross-section for (

**b**) the center NW and (

**c**) a corner NW of a finite N × N array. In (

**b**) and (

**c**), we show also ${\sigma}_{\mathrm{abs}}$ of the infinite periodic array as well as ${P}^{2}$, which is the unit cell area and hence the upper limit for ${\sigma}_{\mathrm{abs}}$ in the infinite periodic array.

**Figure 3.**I

_{sc}in NWs of a finite N × N array. We show here the value for the center NW in the finite array (red circles), the average of all the NWs in the array (magenta circles), an NW at the center of one of the four edges of the array (grey diamonds) and an NW at the corner of the array (blue squares); there are four equivalent edge-centers and corners in the symmetric square array. For N = 1, there is only one NW in the array and these four values therefore coincide. We show also the corresponding value for an NW in the corresponding infinite periodic array (dashed black line).

**Figure 4.**${\left|E\right|}^{2}$ in the x-z plane through the center of the missing nanowire in (

**a**,

**c**) as well as corresponding results for the array without the missing NW in (

**b**,

**d**). (

**a**,

**b**) are for $\lambda =500$ nm and (

**c**,

**d**) are for $\lambda =900$ nm. Here, we show results for x-polarized light; that is, for light with the incident electric field parallel to the x-direction. For the incident electric field, ${\left|{E}_{\mathrm{inc}}\right|}^{2}=1$ [(V/m)

^{2}]. Note that the spatially resolved absorption in the NWs is proportional to ${\left|E\right|}^{2}$ (see, for example, [15]).

**Figure 5.**The compensation of the drop in short-circuit current by increased absorption in neighboring NWs in an NW array with a cluster of M × M missing NWs; that is, M

^{2}missing NWs (red squares). We show here also the estimate $4M{I}_{\mathrm{sc},\mathrm{compensate}}/\left({M}^{2}{I}_{\mathrm{sc},\mathrm{inf}}\right)$, obtained from the absorption response of a finite NW array (black circles).

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Anttu, N.
Absorption of Light in Finite Semiconductor Nanowire Arrays and the Effect of Missing Nanowires. *Symmetry* **2021**, *13*, 1654.
https://doi.org/10.3390/sym13091654

**AMA Style**

Anttu N.
Absorption of Light in Finite Semiconductor Nanowire Arrays and the Effect of Missing Nanowires. *Symmetry*. 2021; 13(9):1654.
https://doi.org/10.3390/sym13091654

**Chicago/Turabian Style**

Anttu, Nicklas.
2021. "Absorption of Light in Finite Semiconductor Nanowire Arrays and the Effect of Missing Nanowires" *Symmetry* 13, no. 9: 1654.
https://doi.org/10.3390/sym13091654