Broadband absorption enhancement in elliptical silicon nanowire arrays for photovoltaic applications Yonggang Wu,1,* Zihuan Xia,1 Zhaoming Liang,1 Jian Zhou,1 Hongfei Jiao,1 Hong Cao,2 and Xuefei Qin1 1
School of Physics Science and Engineering, Tongji University, Shanghai 200092, China 2 Shanghai Center for Photovoltaics, Shanghai 201201, China * [email protected]
Abstract: Semiconducting nanowire arrays have emerged as a promising route toward achieving high efficiencies in solar cells. Here we propose a perpendicular elliptical silicon nanowire (PEE-SiNW) array for broadband light absorption in thin film silicon solar cells. Simulation results reveal that light absorption enhancement is originated from the split of the principal modes as well as the excitation of high order modes caused by the asymmetry of the elliptical nanowires and the enhanced mode coupling between adjacent elliptical nanowires attained by the appropriate arrangement of nanowires. An ultimate efficiency of 29.1% is achieved for the optimal PEE-SiNW array, which is 16.4% higher than that of the circular SiNW array with the same fill fraction. ©2014 Optical Society of America OCIS codes: (040.5350) Photovoltaic; (040.6040) Silicon; (310.6628) Subwavelength structures, nanostructures; (350.4238) Nanophotonics and photonic crystals; (130.2790) Guided waves.
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#213441 - $15.00 USD Received 4 Jun 2014; revised 18 Jul 2014; accepted 28 Jul 2014; published 7 Aug 2014 (C) 2014 OSA 25 August 2014 | Vol. 22, No. S5 | DOI:10.1364/OE.22.0A1292 | OPTICS EXPRESS A1292
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1. Introduction Thin film solar cells composed of a silicon nanowire (SiNW) array have attracted great interests owing to their extraordinary properties such as low reflection loss, superior light trapping, and tailorable energy gaps [1–5]. SiNW arrays can decouple the direction of photon absorption and that of charge transportation by incorporating radial p-n junctions [6, 7]. It is recently shown that light absorption within a semiconducting nanowire array strongly depends on the wire size, geometry, and orientation [8–10], as well as the relative position of the wires and the period of the array [11–13]. A plethora of nanowire arrays composed of disorder radius, position, and length have demonstrated the ability to trap solar lights and to enhance absorption [14–16]. However, few studies have been carried out on arrays composed of nanowires other than circular ones. In this article, we propose a perpendicular elliptical silicon nanowire (PEE-SiNW) array for broadband absorption in thin film silicon solar cells. By adopting elliptical cross section nanowire, the array may possess the advantages derived from both the small and large cross section circular nanowires, which are the efficient antireflection in the short wavelength region and the excitation of numerous resonance modes in the long one [12, 17]. Moreover,
#213441 - $15.00 USD Received 4 Jun 2014; revised 18 Jul 2014; accepted 28 Jul 2014; published 7 Aug 2014 (C) 2014 OSA 25 August 2014 | Vol. 22, No. S5 | DOI:10.1364/OE.22.0A1292 | OPTICS EXPRESS A1293
by placing the nanowires parallel to each other with the major or minor axes of the nearest elliptical nanowires oriented perpendicular to each other, and tuning the rotation angle of the nanowires around their axes, we expect that the surface distance between the elliptical nanowires can be altered and the mode coupling between the nanowires can be optimized. In addition, the reduced transport distance for charge collection along the minor axis of an elliptical nanowire might be advantageous in comparison with a circular one. With the aid of electron beam lithography, reactive ion etching, and holographic lithography, it is possible to control the cross section shape of the nanowires in an array [18–20]. 2. Proposed structure and simulation strategy In our model, the SiNWs are surrounded by air and the PEE-SiNW array is suspended without a substrate as shown schematically in Fig. 1. The major axis of each elliptical nanowire is perpendicular to that of the four nearest ones. The parallel elliptical (PAE) SiNW array, in which the major axes of all the elliptical nanowires are parallel to each other, is shown in Fig. 1 for comparison. The unit cell of the PAE- and PEE-SiNW arrays is enclosed by red dashed lines as shown in Figs. 1(b) and 1(d), respectively. It should be emphasized that due to the introduction of the PEE arrangement, the unit cell of the PEE-SiNW array has quadrupled compared with that of the PAE-SiNW array. The height of the nanowires is set to the standard thin film photovoltaic thickness of 2.33 μm, which is equal to the active layer thickness of the thin film silicon solar cell [12, 21]. The lattice spacing a is chosen to be 600 nm, as the maximum absorption has been observed when circular SiNW arrays have a period of 600 nm [11, 12]. Wavelength-dependent refractive index for c-Si is adopted [22]. The array is illuminated normally by sunlight from the top, unless mentioned otherwise. The Rsoft DiffractMod module based on rigorous coupled wave analysis (RCWA) is used in the simulation. Average absorption over x and y polarized incident lights is adopted. The ultimate efficiency η of the nanowire arrays is evaluated using
η=
λg
300nm
I (λ )A(λ )
4000nm
300nm
λ dλ λg
I (λ )d λ
,
(1)
where I(λ) is the solar energy density at AM1.5d [23], A(λ) the absorptance, and λg the wavelength corresponding to the band gap of silicon. We have verified that the ultimate efficiency of a silicon thin film calculated using the Rsoft RCWA module agrees with that calculated using the analytical formula within 0.3%. When using Eq. (1), all the generated charge carriers are assumed to be collected and contribute to the photocurrent without recombination losses.
#213441 - $15.00 USD Received 4 Jun 2014; revised 18 Jul 2014; accepted 28 Jul 2014; published 7 Aug 2014 (C) 2014 OSA 25 August 2014 | Vol. 22, No. S5 | DOI:10.1364/OE.22.0A1292 | OPTICS EXPRESS A1294
Fig. 1. Schematic of the (a)–(b) PEE- and (c)–(d) PAE-SiNW arrays. (a), (c) pespective, and (b), (d) cross section view. dma and dmi are the diameter of the major and minor axis respectively, and θ is the angle between the major axes and the x or y axis. The unit cell of the two arrays is enclosed by red dashed lines.
3. Results and discussions We first investigate the PEE-SiNW array with the rotation angle θ = 0°. Figure 2 shows the ultimate efficiency of the PEE-SiNW array as a function of the major and minor axis. Symbols A and C denote the positions at which the elliptical and circular SiNW arrays have the maximum ultimate efficiency, respectively, while symbol B denotes the position at which the circular nanowire array has the same fill fraction as that at A. The ultimate efficiency at A, B, and C is 28.2%, 25.0%, and 25.4%, respectively. We notice that, for identical fill fractions between 0.25 and 0.65, the introduction of elliptical nanowires, i.e., moving off the hypotenuse dma = dmi, enhances the efficiency. This indicates that the effect of introducing elliptical nanowires is universal and does not depend on the fill fraction. We also calculate the ultimate efficiency of the PAE-SiNW array as a function of the major and minor axis and find that the maximum ultimate efficiency is 26.3% and the corresponding major and minor axis of the elliptical nanowires are 550 and 375 nm respectively.
#213441 - $15.00 USD Received 4 Jun 2014; revised 18 Jul 2014; accepted 28 Jul 2014; published 7 Aug 2014 (C) 2014 OSA 25 August 2014 | Vol. 22, No. S5 | DOI:10.1364/OE.22.0A1292 | OPTICS EXPRESS A1295
Fig. 2. Ultimate efficiency as a function of the major and minor axis of the PEE-SiNW array. The dashed lines are the contours of the constant fill fraction f ≡ πdmadmi/4a2 for the SiNW arrays. The major and minor axis of the SiNW array at A are 550 and 375 nm, and the diameter of the SiNW array at B and C is 454 and 550 nm, respectively.
The superior performance of the PEE-SiNW array to that of the circular ones in different wavelength regions can clearly be seen in the absorption spectra. Figure 3 displays the absorption spectra of the three arrays corresponding to A, B, and C in Fig. 2. In the wavelength region of 300 – 660 nm, the absorption of the circular SiNW array composed of 550 nm diameter nanowires (array corresponding to C) is weaker than that composed of 454 nm diameter nanowires (array corresponding to B). On the contrary, in the wavelength region of 660 – 1100 nm, the absorption of the latter is smaller than that of the former. This can be attributed to the difference in the fill fraction of the two arrays. The larger fill fraction associated with the former increases the reflection in the short wavelengths due to the high effect refractive index of the SiNW array, while the smaller fill fraction associated with the latter reduces the absorption in the long wavelengths due to the reduction of resonance modes [12, 17]. The PEE-SiNW array possesses distinctively different characteristics than the two circular nanowire arrays. In the short wavelength region, its absorption is equal or higher than that of the circular one with the same fill fraction, while in the long wavelength region, its absorption is higher than that of the cirlular one with larger fill fraction. Therefore, the PEESiNW array exhibits higher absorption not only in the short wavelength region but also in the long one.
Fig. 3. Absorption spectra corresponding to the structures indicated by A, B, and C in Fig. 2. The incident light is normal and unpolarized. The thin lines are the calculated spectra of the arrays, while the thick ones are the smoothed results of the calculated spectra.
#213441 - $15.00 USD Received 4 Jun 2014; revised 18 Jul 2014; accepted 28 Jul 2014; published 7 Aug 2014 (C) 2014 OSA 25 August 2014 | Vol. 22, No. S5 | DOI:10.1364/OE.22.0A1292 | OPTICS EXPRESS A1296
To analyze the electromagnetic wave propagation modes in the circular, PAE-, and PEESiNW arrays, we illustrate the absorption as a function of the wavelength and the nanowire diameter (or major axis) of the three arrays in Fig. 4. In the simulation, the ratio of the major and minor axis is fixed at dma/dmi = 550/375 for the elliptical nanowire arrays. For comparison, the analytical solutions of HE modes for the circular nanowire and eHE and oHE modes for the elliptical nanowire are also shown in Fig. 4. The pre-subscripts e and o denote even and odd respectively. Yeh provided the analytical and theoretical treatment for the propagation of the dominant modes on an elliptical dielectric waveguide. Here we implement the equation provided by Yeh for the calculation of the dispersion relations of HE and eHE1m mode in the silicon nanowire [24] Ce '1 (γ 12 ) γ 12 Fek '1 (γ 02 ) Se '1 (γ 12 ) n02 γ 12 Gek '1 (γ 02 ) + 2 + 2 2 2 2 2 2 Ce1 (γ 1 ) γ 0 Fek1 (γ 0 ) Se1 (γ 1 ) n1 γ 0 Gek1 (γ 0 ) ∞ ∞ χ r ,1α1, r ν s ,1 β1, s γ 2 2 β r =1 s =1 1+ 1 + 2 2 =0 2 α1,1 β1,1 k0 n1 γ0
(2)
2
with 2π
2π
0
0
α m , r = cem (η , −γ 02 )cer (η , γ 12 )dη / cer2 (η , γ 12 )dη , 2π
2π
0
0
β m , r = sem (η , −γ 02 )ser (η , γ 12 )dη / ser2 (η , γ 12 )dη , 2π
2π
0
0
χ r , n = ce 'r (η , γ 12 )sen (η , γ 12 )dη / sen2 (η , γ 12 )dη , 2π
2π
0
0
(3)
ν r , n = se 'r (η , γ 12 )cen (η , γ 12 )dη / cer2 (η , γ 12 )dη . and
γ 12 = (rma2 − rmi2 )(k02 n12 − β 2 ) / 4, γ 02 = (rma2 − rmi2 )( β 2 − k02 n02 ) / 4.
(4)
Here rma and rmi are the semi major and minor axis of the elliptical nanowire respectively. The wave vector k0 equals to 2π/λ where λ is the wavelength in free space. β is the propagation constant, and n1 and n0 are the refractive index of the nanowire and the surrounding medium respectively. cem and sem are the even and odd angular Mathieu function; Ce1 and Se1 are the first kind even and odd radial Mathieu function; Fek1 and Gek1 are the first kind even and odd radial Hankel function, respectively. The prime denotes the derivatives with respect to one of the two elliptical coordinates (ξ,η), as the case may be. For the oHE1m mode calculation, the following parameter swap is implemented: Ce1 ↔ Se1 ,
Ce '1 ↔ Se '1 ,
Fek1 ↔ Gek1 ,
Fek '1 ↔ Gek '1 ,
α1, n ↔ β1, n ,
(5)
ν 1, n ↔ χ1, n .
When a guided wave picks up a round-trip phase change, longitudinal resonances form. Therefore, β is chosen as sπ/L, where L is the length of the nanowire, and s a positive integer. Comparing Figs. 4(b) and 4(c) with Fig. 4(a) in the wavelength between 400 and 600 nm, it is clear that, the principal absorption peaks of the circular SiNW array are distinct and the simulated results are consistent very well with the analytical solutions, e.g., at the positions marked by white dots in Fig. 4(a), indicating that the principal modes referred here as HE
#213441 - $15.00 USD Received 4 Jun 2014; revised 18 Jul 2014; accepted 28 Jul 2014; published 7 Aug 2014 (C) 2014 OSA 25 August 2014 | Vol. 22, No. S5 | DOI:10.1364/OE.22.0A1292 | OPTICS EXPRESS A1297
modes, are confined in the nanowires and the mode coupling between the nanowires is weak. By contrast, the principal absorption peaks of the PAE- and PEE-SiNW arrays become indistinct and the simulated results show a certain deviation from the analytical solutions. Analyzing light polarizations separately for the PAE-SiNW array reveals that the peaks become indistinct due to the polarization mixing. As for the PEE-SiNW array, the indistinct peaks can be attributed to the average absorption over the two perpendicular elliptical nanowire sublattices of the array. It is also evident that, due to the split of the modes, the number of the absorption peaks is greater for the elliptical SiNW arrays than that for the circular one. In the circular cylinder, the principal modes are the double degenerated HE modes, whereas in the elliptical one, the degenerated HE modes split into eHE and oHE modes owing to the asymmetry of the elliptical cylinder. In addition, new absorption peaks, denoted by b7 in Fig. 4(b) and c7 and c8 in Fig. 4(c), emerge, which might be due to the non-weakly guiding effect and will be discussed below.
Fig. 4. Absorption as a function of the wavelength and nanowire diameter (or major axis) of the (a) circular, (b) PAE-, and (c) PEE-SiNW arrays. The black lines in (a) correspond to analytical solutions for the HE1m modes, and the dashed and dot-dashed lines in (b) and (c) to analytical solutions for the eHE1m and oHE1m modes, respectively. The pre-subscripts o and e refer to odd and even respectively. The vertical dashed lines denote the structures with the fill fraction of 0.45.
For the wavelengths longer than 600 nm, many weak absorption peaks appear in addition to those of the HE modes, while for the wavelengths shorter than 600 nm, only a few appears on the intrinsic optical absorption background of silicon. The former can be attributed to the excitation of both the Fabry-Perot modes and Bloch modes, while the latter to the excitation of only the Fabry-Perot modes. According to the theory of waveguide, the circular, PAE-, and
#213441 - $15.00 USD Received 4 Jun 2014; revised 18 Jul 2014; accepted 28 Jul 2014; published 7 Aug 2014 (C) 2014 OSA 25 August 2014 | Vol. 22, No. S5 | DOI:10.1364/OE.22.0A1292 | OPTICS EXPRESS A1298
PEE-SiNW arrays can be regarded as a plane waveguide with thickness of 2.33 μm, and the condition for allowed guided mode propagation is λ ≥ an0/q with q being the diffraction order [25, 26]. Since the periods of the circular and PAE-SiNW arrays are a and that of the PEESiNW array is 2a, light wavelength longer than 600 nm is a prerequisite for the existence of the 1st diffraction order waveguide mode in the circular and PAE-SiNW arrays and the 2nd diffraction order waveguide mode in the PEE-SiNW array. Additionally, the decrease of reflection due to the reflective diffraction threshold in air also contributes to the increase of the absorption in the wavelength longer than 600 nm. As a result, an obvious cut-off appears at λ = 600 nm as shown in Figs. 4(a)–4(c). The number of absorption peaks of the nanowire arrays increases with the diameter or the major axis of the nanowire arrays, which can be attributed to the increase in the effective dielectric coefficient of the nanowire arrays. It is evident from Fig. 4 that, when the fill fraction is 0.45, as denoted by the vertical dashed lines, and the wavelengths are longer than 600 nm, the number of resonance absorption peaks increases when the array evolves from circular to PAE and to PEE. This should be originated from factors other than the change of the fill fraction. In fact, the only variation in the array from the circular to the PAE is the change of the shape of the nanowire, while that from the PAE to the PEE is the change of the arrangement of the nanowires. Therefore, there are reasons to believe that the increase in the absorption peaks in the PEESiNW array is caused by the split of the principal modes due to the introduction of asymmetry and the enhanced mode coupling between the adjacent nanowires due to the adoption of the perpendicular arrangement of the nanowires. In addition, band folding effect introduced by the doubling in the super cell size for the PEE array could also lead to this observed phenomenon. The increase in the number of the absorption peaks can be explained by the energy band diagrams calculated using the plane wave expansion method [27]. Figure 5 shows the dispersion characteristics of the three arrays. It is obvious from Fig. 5 that, the Bloch modes increase when the array evolves from circular to PAE and to PEE, which is consistent with what is observed in Fig. 4. The available Bloch modes whose symmetry matches that of the incident light allow it to be coupled, while flat dispersion curves ensure long lifetime of the Bloch modes. The great number of bands, along with the relatively small group velocities, ensures the broadband light absorption in the PEE-SiNW array [28].
Fig. 5. Dispersion relations of the (a) circular, (b) PAE, and (c) PEE nanowire arrays in the direction of the nanowire axis. The fill fraction of all the three arrays is 0.45. The dielectric coefficient of the nanowires is chosen as 13.69, while the major and minor axis of the elliptical nanowire are 550 and 375 nm respectively. Dashed lines are the dispersion relations of light in homogeneous silicon material.
#213441 - $15.00 USD Received 4 Jun 2014; revised 18 Jul 2014; accepted 28 Jul 2014; published 7 Aug 2014 (C) 2014 OSA 25 August 2014 | Vol. 22, No. S5 | DOI:10.1364/OE.22.0A1292 | OPTICS EXPRESS A1299
Figure 6 illustrates the magnetic field patterns of the circular and elliptical SiNW arrays corresponding to the positions denoted by the characters in Figs. 4(a)–4(c). The magnetic field distributions are calculated by the Rsoft DiffractMod module based on RCWA. For the circular nanowires, the magnetic fields at a1, a2, and a3 (Fig. 6(a)) correspond to HE11, HE12, and HE13 modes, respectively. Only the HE1m modes and not any of the other TE, TM, EH modes contribute to the absorption in the circular nanowire array as shown in Fig. 4(a). This is because the former are the only modes with the same angular quantization number as and the similar electric field patterns to that of the incident plane wave outside the SiNW core [29, 30]. When the nanowires evolve from circular to elliptical, the HE1m modes split into oHE1m and eHE1m modes as shown in Figs. 6(b) and 6(c). From Fig. 6(c), we notice that the oHE1m and eHE1m modes correspond to the polarizations along the major and minor axis respectively. The electromagnetic field power concentrates in the diagonal sublattices. It is interesting that the field distribution patterns shown in Figs. 6(b7), 6(c7), and 6(c8) are similar to none of those obtained for the weakly guiding elliptical dielectric waveguides [31, 32]. This might be attributed to the adoption of non-weakly guiding silicon nanowires in our model, which results in the emergence of new modes in addition to those obtained through the weakly guiding approximation. We notice that the field distribution patterns at positions b7, c7, and c8 in Fig. 4 are to a certain degree similar to those at the neighbours of these positions, e.g., the field distribution of Fig. 6(c8) on the x axis is identical to those of Figs. 6(c5) and 6(c6) on the same axis, therefore the modes corresponding to b7, c7, and c8 are referred to as the modified HE13, HE12, HE13, respectively.
Fig. 6. Magnetic field distribution of the circular and elliptical nanowire arrays corresponding to the positions denoted by the characters in Fig. 4. The patterns b1, b3, and b5 are for the x component of the magnetic field illuminated by light with its electric field polarized in y direction, while the others are for the y component of the magnetic field illuminated by light with its electric field polarized in x direction.
#213441 - $15.00 USD Received 4 Jun 2014; revised 18 Jul 2014; accepted 28 Jul 2014; published 7 Aug 2014 (C) 2014 OSA 25 August 2014 | Vol. 22, No. S5 | DOI:10.1364/OE.22.0A1292 | OPTICS EXPRESS A1300
Encouraged by the improvement of the light absorption of the PEE-SiNW array discussed above over that of the PAE-SiNW arrays, we calculate the ultimate efficiency of the PEE- and PAE-SiNW arrays as a function of the rotation angle θ with the anticipation that an appropriate nanowire arrangement can improve the mode coupling between the adjacent nanowires and might further increase the light absorption. The maximum ultimate efficiency of 29.1% and 26.5% is obtained for the PEE-SiNW array with a rotation angle of 25° and the PAE-SiNW array with a rotation angle of 10°, respectively, as shown in Fig. 7. They are 16.4% and 6.0% higher than that of the circular SiNW array with the same fill fraction. Cao et al. have demonstrated that broadband absorption can be achieved through efficient optical couplings between two circular nanowires [33], and Sturmberg et al. have pointed out that distances between nanowires play an important part in the absorption of a nanowire array [34]. These effects are also observed in the elliptical SiNW arrays. In addition, we have observed that for both the PAE and PEE array, structures without lattice symmetry in general absorb more light than those with a higher lattice symmetry (mirror symmetry and/or 90° rotational symmetry), and rotating the angle of the PAE and PEE array can affect the excitation of the resonance modes, in a large part, by tuning the underlying lattice symmetry.
Fig. 7. Ultimate efficiency of the PAE- and PEE-SiNW arrays as a function of the rotation angle of the nanowires. The rotation angles of the nanowires for three typical positions: (a) and (d) 0°, (b) and (e) 25°, and (c) and (f) 45° are schematically depicted in the insets.
For practical applications of solar cells, their ability to absorb light over a wide incident angle range is crucial, which not only contributes to the integrated power generation but also eliminates the employment of solar tracking units. Therefore, we investigate the dependence of ultimate efficiency on the incident angle of light for the circular and elliptical nanowire arrays. The result is shown in Fig. 8. In the incident angle range 0 – 85°, the ultimate efficiency of the PAE-SiNW arrays is greater than that of the circular array. In the incident angle range 0 – 70°, the PEE-SiNW arrays displays superior performance to both the circular and PAE nanowire arrays while the one with a 25° rotation exhibits the best performance. The ultimate efficiency of the nanowire arrays increases with the incident angle when it is less than 20° for the circular and PAE-SiNW arrays and 10° for the PEE-SiNW array with a 0° rotation, which might be due to the excitation of anti-symmetrical modes and the increase in the path length of light in the arrays [35].
#213441 - $15.00 USD Received 4 Jun 2014; revised 18 Jul 2014; accepted 28 Jul 2014; published 7 Aug 2014 (C) 2014 OSA 25 August 2014 | Vol. 22, No. S5 | DOI:10.1364/OE.22.0A1292 | OPTICS EXPRESS A1301
Fig. 8. Ultimate efficiency of the circular and elliptical nanowire arrays as a function of the incident angle of the light. The five nanowire arrays have the identical fill fraction of 0.45, and the lattice spacing of 600 nm. The ratio of the major and minor axis of the elliptical nanowire is 550/375.
4. Conclusions PEE-SiNW arrays possess superior broadband and broad incident angle characteristics to that of circular arrays. The split of the principal modes and the excitation of high order modes due to the asymmetry in elliptical nanowires and the appropriate rotation of the nanowires around their axes introduce more resonance modes and thus contribute to the light absorption enhancement of the PEE-SiNW array. An ultimate efficiency of 28.2% is achieved for the PEE-SiNW array with a 0° nanowire rotation angle. It is further increased to 29.1% when the rotation angle is 25°, which is 16.4% higher than that of the circular SiNW array with the same fill fraction. Acknowledgments This research is sponsored by the National Natural Science Foundation of China (No. 60977028) and Key Project Foundation of Shanghai (No. 09JC1413800).
#213441 - $15.00 USD Received 4 Jun 2014; revised 18 Jul 2014; accepted 28 Jul 2014; published 7 Aug 2014 (C) 2014 OSA 25 August 2014 | Vol. 22, No. S5 | DOI:10.1364/OE.22.0A1292 | OPTICS EXPRESS A1302
School of Physics Science and Engineering, Tongji University, Shanghai 200092, China 2 Shanghai Center for Photovoltaics, Shanghai 201201, China * [email protected]
Abstract: Semiconducting nanowire arrays have emerged as a promising route toward achieving high efficiencies in solar cells. Here we propose a perpendicular elliptical silicon nanowire (PEE-SiNW) array for broadband light absorption in thin film silicon solar cells. Simulation results reveal that light absorption enhancement is originated from the split of the principal modes as well as the excitation of high order modes caused by the asymmetry of the elliptical nanowires and the enhanced mode coupling between adjacent elliptical nanowires attained by the appropriate arrangement of nanowires. An ultimate efficiency of 29.1% is achieved for the optimal PEE-SiNW array, which is 16.4% higher than that of the circular SiNW array with the same fill fraction. ©2014 Optical Society of America OCIS codes: (040.5350) Photovoltaic; (040.6040) Silicon; (310.6628) Subwavelength structures, nanostructures; (350.4238) Nanophotonics and photonic crystals; (130.2790) Guided waves.
References and links 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.
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#213441 - $15.00 USD Received 4 Jun 2014; revised 18 Jul 2014; accepted 28 Jul 2014; published 7 Aug 2014 (C) 2014 OSA 25 August 2014 | Vol. 22, No. S5 | DOI:10.1364/OE.22.0A1292 | OPTICS EXPRESS A1292
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1. Introduction Thin film solar cells composed of a silicon nanowire (SiNW) array have attracted great interests owing to their extraordinary properties such as low reflection loss, superior light trapping, and tailorable energy gaps [1–5]. SiNW arrays can decouple the direction of photon absorption and that of charge transportation by incorporating radial p-n junctions [6, 7]. It is recently shown that light absorption within a semiconducting nanowire array strongly depends on the wire size, geometry, and orientation [8–10], as well as the relative position of the wires and the period of the array [11–13]. A plethora of nanowire arrays composed of disorder radius, position, and length have demonstrated the ability to trap solar lights and to enhance absorption [14–16]. However, few studies have been carried out on arrays composed of nanowires other than circular ones. In this article, we propose a perpendicular elliptical silicon nanowire (PEE-SiNW) array for broadband absorption in thin film silicon solar cells. By adopting elliptical cross section nanowire, the array may possess the advantages derived from both the small and large cross section circular nanowires, which are the efficient antireflection in the short wavelength region and the excitation of numerous resonance modes in the long one [12, 17]. Moreover,
#213441 - $15.00 USD Received 4 Jun 2014; revised 18 Jul 2014; accepted 28 Jul 2014; published 7 Aug 2014 (C) 2014 OSA 25 August 2014 | Vol. 22, No. S5 | DOI:10.1364/OE.22.0A1292 | OPTICS EXPRESS A1293
by placing the nanowires parallel to each other with the major or minor axes of the nearest elliptical nanowires oriented perpendicular to each other, and tuning the rotation angle of the nanowires around their axes, we expect that the surface distance between the elliptical nanowires can be altered and the mode coupling between the nanowires can be optimized. In addition, the reduced transport distance for charge collection along the minor axis of an elliptical nanowire might be advantageous in comparison with a circular one. With the aid of electron beam lithography, reactive ion etching, and holographic lithography, it is possible to control the cross section shape of the nanowires in an array [18–20]. 2. Proposed structure and simulation strategy In our model, the SiNWs are surrounded by air and the PEE-SiNW array is suspended without a substrate as shown schematically in Fig. 1. The major axis of each elliptical nanowire is perpendicular to that of the four nearest ones. The parallel elliptical (PAE) SiNW array, in which the major axes of all the elliptical nanowires are parallel to each other, is shown in Fig. 1 for comparison. The unit cell of the PAE- and PEE-SiNW arrays is enclosed by red dashed lines as shown in Figs. 1(b) and 1(d), respectively. It should be emphasized that due to the introduction of the PEE arrangement, the unit cell of the PEE-SiNW array has quadrupled compared with that of the PAE-SiNW array. The height of the nanowires is set to the standard thin film photovoltaic thickness of 2.33 μm, which is equal to the active layer thickness of the thin film silicon solar cell [12, 21]. The lattice spacing a is chosen to be 600 nm, as the maximum absorption has been observed when circular SiNW arrays have a period of 600 nm [11, 12]. Wavelength-dependent refractive index for c-Si is adopted [22]. The array is illuminated normally by sunlight from the top, unless mentioned otherwise. The Rsoft DiffractMod module based on rigorous coupled wave analysis (RCWA) is used in the simulation. Average absorption over x and y polarized incident lights is adopted. The ultimate efficiency η of the nanowire arrays is evaluated using
η=
λg
300nm
I (λ )A(λ )
4000nm
300nm
λ dλ λg
I (λ )d λ
,
(1)
where I(λ) is the solar energy density at AM1.5d [23], A(λ) the absorptance, and λg the wavelength corresponding to the band gap of silicon. We have verified that the ultimate efficiency of a silicon thin film calculated using the Rsoft RCWA module agrees with that calculated using the analytical formula within 0.3%. When using Eq. (1), all the generated charge carriers are assumed to be collected and contribute to the photocurrent without recombination losses.
#213441 - $15.00 USD Received 4 Jun 2014; revised 18 Jul 2014; accepted 28 Jul 2014; published 7 Aug 2014 (C) 2014 OSA 25 August 2014 | Vol. 22, No. S5 | DOI:10.1364/OE.22.0A1292 | OPTICS EXPRESS A1294
Fig. 1. Schematic of the (a)–(b) PEE- and (c)–(d) PAE-SiNW arrays. (a), (c) pespective, and (b), (d) cross section view. dma and dmi are the diameter of the major and minor axis respectively, and θ is the angle between the major axes and the x or y axis. The unit cell of the two arrays is enclosed by red dashed lines.
3. Results and discussions We first investigate the PEE-SiNW array with the rotation angle θ = 0°. Figure 2 shows the ultimate efficiency of the PEE-SiNW array as a function of the major and minor axis. Symbols A and C denote the positions at which the elliptical and circular SiNW arrays have the maximum ultimate efficiency, respectively, while symbol B denotes the position at which the circular nanowire array has the same fill fraction as that at A. The ultimate efficiency at A, B, and C is 28.2%, 25.0%, and 25.4%, respectively. We notice that, for identical fill fractions between 0.25 and 0.65, the introduction of elliptical nanowires, i.e., moving off the hypotenuse dma = dmi, enhances the efficiency. This indicates that the effect of introducing elliptical nanowires is universal and does not depend on the fill fraction. We also calculate the ultimate efficiency of the PAE-SiNW array as a function of the major and minor axis and find that the maximum ultimate efficiency is 26.3% and the corresponding major and minor axis of the elliptical nanowires are 550 and 375 nm respectively.
#213441 - $15.00 USD Received 4 Jun 2014; revised 18 Jul 2014; accepted 28 Jul 2014; published 7 Aug 2014 (C) 2014 OSA 25 August 2014 | Vol. 22, No. S5 | DOI:10.1364/OE.22.0A1292 | OPTICS EXPRESS A1295
Fig. 2. Ultimate efficiency as a function of the major and minor axis of the PEE-SiNW array. The dashed lines are the contours of the constant fill fraction f ≡ πdmadmi/4a2 for the SiNW arrays. The major and minor axis of the SiNW array at A are 550 and 375 nm, and the diameter of the SiNW array at B and C is 454 and 550 nm, respectively.
The superior performance of the PEE-SiNW array to that of the circular ones in different wavelength regions can clearly be seen in the absorption spectra. Figure 3 displays the absorption spectra of the three arrays corresponding to A, B, and C in Fig. 2. In the wavelength region of 300 – 660 nm, the absorption of the circular SiNW array composed of 550 nm diameter nanowires (array corresponding to C) is weaker than that composed of 454 nm diameter nanowires (array corresponding to B). On the contrary, in the wavelength region of 660 – 1100 nm, the absorption of the latter is smaller than that of the former. This can be attributed to the difference in the fill fraction of the two arrays. The larger fill fraction associated with the former increases the reflection in the short wavelengths due to the high effect refractive index of the SiNW array, while the smaller fill fraction associated with the latter reduces the absorption in the long wavelengths due to the reduction of resonance modes [12, 17]. The PEE-SiNW array possesses distinctively different characteristics than the two circular nanowire arrays. In the short wavelength region, its absorption is equal or higher than that of the circular one with the same fill fraction, while in the long wavelength region, its absorption is higher than that of the cirlular one with larger fill fraction. Therefore, the PEESiNW array exhibits higher absorption not only in the short wavelength region but also in the long one.
Fig. 3. Absorption spectra corresponding to the structures indicated by A, B, and C in Fig. 2. The incident light is normal and unpolarized. The thin lines are the calculated spectra of the arrays, while the thick ones are the smoothed results of the calculated spectra.
#213441 - $15.00 USD Received 4 Jun 2014; revised 18 Jul 2014; accepted 28 Jul 2014; published 7 Aug 2014 (C) 2014 OSA 25 August 2014 | Vol. 22, No. S5 | DOI:10.1364/OE.22.0A1292 | OPTICS EXPRESS A1296
To analyze the electromagnetic wave propagation modes in the circular, PAE-, and PEESiNW arrays, we illustrate the absorption as a function of the wavelength and the nanowire diameter (or major axis) of the three arrays in Fig. 4. In the simulation, the ratio of the major and minor axis is fixed at dma/dmi = 550/375 for the elliptical nanowire arrays. For comparison, the analytical solutions of HE modes for the circular nanowire and eHE and oHE modes for the elliptical nanowire are also shown in Fig. 4. The pre-subscripts e and o denote even and odd respectively. Yeh provided the analytical and theoretical treatment for the propagation of the dominant modes on an elliptical dielectric waveguide. Here we implement the equation provided by Yeh for the calculation of the dispersion relations of HE and eHE1m mode in the silicon nanowire [24] Ce '1 (γ 12 ) γ 12 Fek '1 (γ 02 ) Se '1 (γ 12 ) n02 γ 12 Gek '1 (γ 02 ) + 2 + 2 2 2 2 2 2 Ce1 (γ 1 ) γ 0 Fek1 (γ 0 ) Se1 (γ 1 ) n1 γ 0 Gek1 (γ 0 ) ∞ ∞ χ r ,1α1, r ν s ,1 β1, s γ 2 2 β r =1 s =1 1+ 1 + 2 2 =0 2 α1,1 β1,1 k0 n1 γ0
(2)
2
with 2π
2π
0
0
α m , r = cem (η , −γ 02 )cer (η , γ 12 )dη / cer2 (η , γ 12 )dη , 2π
2π
0
0
β m , r = sem (η , −γ 02 )ser (η , γ 12 )dη / ser2 (η , γ 12 )dη , 2π
2π
0
0
χ r , n = ce 'r (η , γ 12 )sen (η , γ 12 )dη / sen2 (η , γ 12 )dη , 2π
2π
0
0
(3)
ν r , n = se 'r (η , γ 12 )cen (η , γ 12 )dη / cer2 (η , γ 12 )dη . and
γ 12 = (rma2 − rmi2 )(k02 n12 − β 2 ) / 4, γ 02 = (rma2 − rmi2 )( β 2 − k02 n02 ) / 4.
(4)
Here rma and rmi are the semi major and minor axis of the elliptical nanowire respectively. The wave vector k0 equals to 2π/λ where λ is the wavelength in free space. β is the propagation constant, and n1 and n0 are the refractive index of the nanowire and the surrounding medium respectively. cem and sem are the even and odd angular Mathieu function; Ce1 and Se1 are the first kind even and odd radial Mathieu function; Fek1 and Gek1 are the first kind even and odd radial Hankel function, respectively. The prime denotes the derivatives with respect to one of the two elliptical coordinates (ξ,η), as the case may be. For the oHE1m mode calculation, the following parameter swap is implemented: Ce1 ↔ Se1 ,
Ce '1 ↔ Se '1 ,
Fek1 ↔ Gek1 ,
Fek '1 ↔ Gek '1 ,
α1, n ↔ β1, n ,
(5)
ν 1, n ↔ χ1, n .
When a guided wave picks up a round-trip phase change, longitudinal resonances form. Therefore, β is chosen as sπ/L, where L is the length of the nanowire, and s a positive integer. Comparing Figs. 4(b) and 4(c) with Fig. 4(a) in the wavelength between 400 and 600 nm, it is clear that, the principal absorption peaks of the circular SiNW array are distinct and the simulated results are consistent very well with the analytical solutions, e.g., at the positions marked by white dots in Fig. 4(a), indicating that the principal modes referred here as HE
#213441 - $15.00 USD Received 4 Jun 2014; revised 18 Jul 2014; accepted 28 Jul 2014; published 7 Aug 2014 (C) 2014 OSA 25 August 2014 | Vol. 22, No. S5 | DOI:10.1364/OE.22.0A1292 | OPTICS EXPRESS A1297
modes, are confined in the nanowires and the mode coupling between the nanowires is weak. By contrast, the principal absorption peaks of the PAE- and PEE-SiNW arrays become indistinct and the simulated results show a certain deviation from the analytical solutions. Analyzing light polarizations separately for the PAE-SiNW array reveals that the peaks become indistinct due to the polarization mixing. As for the PEE-SiNW array, the indistinct peaks can be attributed to the average absorption over the two perpendicular elliptical nanowire sublattices of the array. It is also evident that, due to the split of the modes, the number of the absorption peaks is greater for the elliptical SiNW arrays than that for the circular one. In the circular cylinder, the principal modes are the double degenerated HE modes, whereas in the elliptical one, the degenerated HE modes split into eHE and oHE modes owing to the asymmetry of the elliptical cylinder. In addition, new absorption peaks, denoted by b7 in Fig. 4(b) and c7 and c8 in Fig. 4(c), emerge, which might be due to the non-weakly guiding effect and will be discussed below.
Fig. 4. Absorption as a function of the wavelength and nanowire diameter (or major axis) of the (a) circular, (b) PAE-, and (c) PEE-SiNW arrays. The black lines in (a) correspond to analytical solutions for the HE1m modes, and the dashed and dot-dashed lines in (b) and (c) to analytical solutions for the eHE1m and oHE1m modes, respectively. The pre-subscripts o and e refer to odd and even respectively. The vertical dashed lines denote the structures with the fill fraction of 0.45.
For the wavelengths longer than 600 nm, many weak absorption peaks appear in addition to those of the HE modes, while for the wavelengths shorter than 600 nm, only a few appears on the intrinsic optical absorption background of silicon. The former can be attributed to the excitation of both the Fabry-Perot modes and Bloch modes, while the latter to the excitation of only the Fabry-Perot modes. According to the theory of waveguide, the circular, PAE-, and
#213441 - $15.00 USD Received 4 Jun 2014; revised 18 Jul 2014; accepted 28 Jul 2014; published 7 Aug 2014 (C) 2014 OSA 25 August 2014 | Vol. 22, No. S5 | DOI:10.1364/OE.22.0A1292 | OPTICS EXPRESS A1298
PEE-SiNW arrays can be regarded as a plane waveguide with thickness of 2.33 μm, and the condition for allowed guided mode propagation is λ ≥ an0/q with q being the diffraction order [25, 26]. Since the periods of the circular and PAE-SiNW arrays are a and that of the PEESiNW array is 2a, light wavelength longer than 600 nm is a prerequisite for the existence of the 1st diffraction order waveguide mode in the circular and PAE-SiNW arrays and the 2nd diffraction order waveguide mode in the PEE-SiNW array. Additionally, the decrease of reflection due to the reflective diffraction threshold in air also contributes to the increase of the absorption in the wavelength longer than 600 nm. As a result, an obvious cut-off appears at λ = 600 nm as shown in Figs. 4(a)–4(c). The number of absorption peaks of the nanowire arrays increases with the diameter or the major axis of the nanowire arrays, which can be attributed to the increase in the effective dielectric coefficient of the nanowire arrays. It is evident from Fig. 4 that, when the fill fraction is 0.45, as denoted by the vertical dashed lines, and the wavelengths are longer than 600 nm, the number of resonance absorption peaks increases when the array evolves from circular to PAE and to PEE. This should be originated from factors other than the change of the fill fraction. In fact, the only variation in the array from the circular to the PAE is the change of the shape of the nanowire, while that from the PAE to the PEE is the change of the arrangement of the nanowires. Therefore, there are reasons to believe that the increase in the absorption peaks in the PEESiNW array is caused by the split of the principal modes due to the introduction of asymmetry and the enhanced mode coupling between the adjacent nanowires due to the adoption of the perpendicular arrangement of the nanowires. In addition, band folding effect introduced by the doubling in the super cell size for the PEE array could also lead to this observed phenomenon. The increase in the number of the absorption peaks can be explained by the energy band diagrams calculated using the plane wave expansion method [27]. Figure 5 shows the dispersion characteristics of the three arrays. It is obvious from Fig. 5 that, the Bloch modes increase when the array evolves from circular to PAE and to PEE, which is consistent with what is observed in Fig. 4. The available Bloch modes whose symmetry matches that of the incident light allow it to be coupled, while flat dispersion curves ensure long lifetime of the Bloch modes. The great number of bands, along with the relatively small group velocities, ensures the broadband light absorption in the PEE-SiNW array [28].
Fig. 5. Dispersion relations of the (a) circular, (b) PAE, and (c) PEE nanowire arrays in the direction of the nanowire axis. The fill fraction of all the three arrays is 0.45. The dielectric coefficient of the nanowires is chosen as 13.69, while the major and minor axis of the elliptical nanowire are 550 and 375 nm respectively. Dashed lines are the dispersion relations of light in homogeneous silicon material.
#213441 - $15.00 USD Received 4 Jun 2014; revised 18 Jul 2014; accepted 28 Jul 2014; published 7 Aug 2014 (C) 2014 OSA 25 August 2014 | Vol. 22, No. S5 | DOI:10.1364/OE.22.0A1292 | OPTICS EXPRESS A1299
Figure 6 illustrates the magnetic field patterns of the circular and elliptical SiNW arrays corresponding to the positions denoted by the characters in Figs. 4(a)–4(c). The magnetic field distributions are calculated by the Rsoft DiffractMod module based on RCWA. For the circular nanowires, the magnetic fields at a1, a2, and a3 (Fig. 6(a)) correspond to HE11, HE12, and HE13 modes, respectively. Only the HE1m modes and not any of the other TE, TM, EH modes contribute to the absorption in the circular nanowire array as shown in Fig. 4(a). This is because the former are the only modes with the same angular quantization number as and the similar electric field patterns to that of the incident plane wave outside the SiNW core [29, 30]. When the nanowires evolve from circular to elliptical, the HE1m modes split into oHE1m and eHE1m modes as shown in Figs. 6(b) and 6(c). From Fig. 6(c), we notice that the oHE1m and eHE1m modes correspond to the polarizations along the major and minor axis respectively. The electromagnetic field power concentrates in the diagonal sublattices. It is interesting that the field distribution patterns shown in Figs. 6(b7), 6(c7), and 6(c8) are similar to none of those obtained for the weakly guiding elliptical dielectric waveguides [31, 32]. This might be attributed to the adoption of non-weakly guiding silicon nanowires in our model, which results in the emergence of new modes in addition to those obtained through the weakly guiding approximation. We notice that the field distribution patterns at positions b7, c7, and c8 in Fig. 4 are to a certain degree similar to those at the neighbours of these positions, e.g., the field distribution of Fig. 6(c8) on the x axis is identical to those of Figs. 6(c5) and 6(c6) on the same axis, therefore the modes corresponding to b7, c7, and c8 are referred to as the modified HE13, HE12, HE13, respectively.
Fig. 6. Magnetic field distribution of the circular and elliptical nanowire arrays corresponding to the positions denoted by the characters in Fig. 4. The patterns b1, b3, and b5 are for the x component of the magnetic field illuminated by light with its electric field polarized in y direction, while the others are for the y component of the magnetic field illuminated by light with its electric field polarized in x direction.
#213441 - $15.00 USD Received 4 Jun 2014; revised 18 Jul 2014; accepted 28 Jul 2014; published 7 Aug 2014 (C) 2014 OSA 25 August 2014 | Vol. 22, No. S5 | DOI:10.1364/OE.22.0A1292 | OPTICS EXPRESS A1300
Encouraged by the improvement of the light absorption of the PEE-SiNW array discussed above over that of the PAE-SiNW arrays, we calculate the ultimate efficiency of the PEE- and PAE-SiNW arrays as a function of the rotation angle θ with the anticipation that an appropriate nanowire arrangement can improve the mode coupling between the adjacent nanowires and might further increase the light absorption. The maximum ultimate efficiency of 29.1% and 26.5% is obtained for the PEE-SiNW array with a rotation angle of 25° and the PAE-SiNW array with a rotation angle of 10°, respectively, as shown in Fig. 7. They are 16.4% and 6.0% higher than that of the circular SiNW array with the same fill fraction. Cao et al. have demonstrated that broadband absorption can be achieved through efficient optical couplings between two circular nanowires [33], and Sturmberg et al. have pointed out that distances between nanowires play an important part in the absorption of a nanowire array [34]. These effects are also observed in the elliptical SiNW arrays. In addition, we have observed that for both the PAE and PEE array, structures without lattice symmetry in general absorb more light than those with a higher lattice symmetry (mirror symmetry and/or 90° rotational symmetry), and rotating the angle of the PAE and PEE array can affect the excitation of the resonance modes, in a large part, by tuning the underlying lattice symmetry.
Fig. 7. Ultimate efficiency of the PAE- and PEE-SiNW arrays as a function of the rotation angle of the nanowires. The rotation angles of the nanowires for three typical positions: (a) and (d) 0°, (b) and (e) 25°, and (c) and (f) 45° are schematically depicted in the insets.
For practical applications of solar cells, their ability to absorb light over a wide incident angle range is crucial, which not only contributes to the integrated power generation but also eliminates the employment of solar tracking units. Therefore, we investigate the dependence of ultimate efficiency on the incident angle of light for the circular and elliptical nanowire arrays. The result is shown in Fig. 8. In the incident angle range 0 – 85°, the ultimate efficiency of the PAE-SiNW arrays is greater than that of the circular array. In the incident angle range 0 – 70°, the PEE-SiNW arrays displays superior performance to both the circular and PAE nanowire arrays while the one with a 25° rotation exhibits the best performance. The ultimate efficiency of the nanowire arrays increases with the incident angle when it is less than 20° for the circular and PAE-SiNW arrays and 10° for the PEE-SiNW array with a 0° rotation, which might be due to the excitation of anti-symmetrical modes and the increase in the path length of light in the arrays [35].
#213441 - $15.00 USD Received 4 Jun 2014; revised 18 Jul 2014; accepted 28 Jul 2014; published 7 Aug 2014 (C) 2014 OSA 25 August 2014 | Vol. 22, No. S5 | DOI:10.1364/OE.22.0A1292 | OPTICS EXPRESS A1301
Fig. 8. Ultimate efficiency of the circular and elliptical nanowire arrays as a function of the incident angle of the light. The five nanowire arrays have the identical fill fraction of 0.45, and the lattice spacing of 600 nm. The ratio of the major and minor axis of the elliptical nanowire is 550/375.
4. Conclusions PEE-SiNW arrays possess superior broadband and broad incident angle characteristics to that of circular arrays. The split of the principal modes and the excitation of high order modes due to the asymmetry in elliptical nanowires and the appropriate rotation of the nanowires around their axes introduce more resonance modes and thus contribute to the light absorption enhancement of the PEE-SiNW array. An ultimate efficiency of 28.2% is achieved for the PEE-SiNW array with a 0° nanowire rotation angle. It is further increased to 29.1% when the rotation angle is 25°, which is 16.4% higher than that of the circular SiNW array with the same fill fraction. Acknowledgments This research is sponsored by the National Natural Science Foundation of China (No. 60977028) and Key Project Foundation of Shanghai (No. 09JC1413800).
#213441 - $15.00 USD Received 4 Jun 2014; revised 18 Jul 2014; accepted 28 Jul 2014; published 7 Aug 2014 (C) 2014 OSA 25 August 2014 | Vol. 22, No. S5 | DOI:10.1364/OE.22.0A1292 | OPTICS EXPRESS A1302
