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The optical absorption of nanowire arrays
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Photonics and Nanostructures – Fundamentals and Applications 9 (2011) 163–167 www.elsevier.com/locate/photonics
The optical absorption of nanowire arrays N. Lagos a,b, M.M. Sigalas a,*, D. Niarchos b a
Institute of Materials Science, N.C.S.R ‘‘Demokritos’’ Agia Paraskevi, 15310 Athens, Greece b Materials Science Department, University of Patras, 26504 Patras, Greece
Received 25 January 2010; received in revised form 16 July 2010; accepted 2 September 2010 Available online 15 September 2010
Abstract The optical absorption of nanowire arrays is calculated with the rigorous coupled wave analysis. The effect of different parameters such as periodicity, filling ratio and thickness are studied. There are potential applications of these arrays for photovoltaic applications and there are experiments on Si nanowire arrays. For that reason the main objective of this study is to find the maximum absorption of nanowire arrays over a frequency area that covers the visible spectrum. There is a random location and orientation of the nanowires in those experiments. For that reason the effect of disorder in the absorption results is also examined. Although, the main focus of this study is Si nanowires, other materials are also calculated. # 2010 Elsevier B.V. All rights reserved. Keywords: Nanowires; Solar cells; Absorption; Disorder
1. Introduction There is a growing interest in photovoltaics (PV) since there is one of the main renewable energy sources. PVs based on thick (100 mm or thicker) silicon materials are still the main commercially used technology. One of the problems with this technology is the high amount of material used which accounts for almost half of the cost of those devices [1]. For that reason, there are intense efforts looking for thin films technologies as well as different materials that can give similar or even higher efficiencies with less material, therefore reduced costs [1]. One of the new directions in PVs is related with nanowire arrays [2–9]. Due to the bottom-up design followed in their fabrication, they can give good control of important parameters that affect the PV performance such as chemical and dopant composition, size and structure of
* Corresponding author. E-mail address: [email protected] (M.M. Sigalas). 1569-4410/$ – see front matter # 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.photonics.2010.09.005
the diode junction [2]. There are two types of nanowires, the axial where the p–i–n junction is formed along their axis and the radial where the p–i–n junction is formed along their radius. The radial nanowires have carrier collection distance smaller or comparable with the minority carrier diffusion length. That reduces significantly losses of the carriers in the bulk and increases the possibility of the photogenerated carriers to reach the contacts [2]. Here, the different parameters that affect the optical absorption are examined. The Rigorous Coupled Wave Analysis (RCWA) is used to calculate the reflection (R) and transmission (T) of electromagnetic waves from nanowire arrays [10,11]. The main goal of this structure is to find the optimum parameters (periodicity, filling ratio, material) that give the maximum absorption (A = 1 T R) for the maximum possible visible frequency area. The effect of the disorder is also studied since in all the fabricated nanowire arrays is always present. The dielectric constants used in the calculations are from measurements [12]. In particular, both Si as
[(Fig._1)TD$IG] 164
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Fig. 1. The xz and xy cross-sections of the nanowire structure.
well as the other dielectric materials studied and Ag have frequency dependent dielectric constants that are taken by fitting the measured results [12]. It should be mentioned that RCWA is a frequency domain method and it needs the dielectric constant in each frequency. 2. Results and discussion The structure used in the calculations is shown in Fig. 1. It is a square lattice with lattice constant, a, and radius of the nanowires, r. The incident light is along the z-axis and polarized along the x-axis. The calculations show that increasing the thickness of the nanowires their absorption also increases in accordance with previous studies [6]. They also show that there are no significant changes in the absorption spectrum for incident angles less that 208 for both polarizations. So, all the following results are for zero incident angle and t = 4 mm thick nanowires. For lattice constant a = 300 nm, the absorption as a function of energy is shown in Fig. 2. The general trend of the absorption resembles a step function. There is high absorption at high energies where Si is highly absorbing and lower absorption at lower energies. The goal of the present study is to find the optimum parameters for which the high absorption area can cover as much as possible of the visible spectrum and if possible some part of the near IR region since these are the areas where most of the sunlight energy is concentrated. One way to achieve that is to increase the radius of the nanowires. For r/a = 0.25, the absorption is almost 100% for energies higher than 2.9 eVand drops below 10% for energies less than 2.5 eV (Fig. 2). Increasing the radius to 0.5a, the absorption stays above 50% for energies above 1.6 eV covering all the visible spectrum. The drop of the absorption at around 2.4 eV is due to the increased scattering from the nanowire arrays that leads to higher reflection. Increasing the radius to 0.75a, the absorption is between 70% and 90% for energies above 2.2 eV and drops below 50% for
energies higher than 1.5 eV similarly as in the r/a = 0.5 case. As a comparison, Fig. 2 also shows the uniform Si case with thickness of 4 mm, where the maximum absorption is 64% at 2.2 eV. The absorption is higher than the uniform Si slab, for most of the visible spectrum for the cases with r/a = 0.5 and 0.75. Increasing the lattice constant, the scattering of the light from the array becomes more significant since the wavelength of light becomes comparable with the feature sizes. That becomes clear for a = 600 nm (Fig. 3), where the absorption spectrum for all radius show significant fluctuations. For r/a = 0.25, the absorption is above 20% but lower than 50%, for all the energies above 1.8 eV. Increasing the radius to 0.5a, increases the absorption at higher energies and significant absorption above 10% appears for energies as low as 0.9 eV. However, the r/a = 0.75 has the best absorption spectrum since it remains above 70% for most of the energies above 1.6 eV covering the whole visible spectrum. Keeping the r/a equal to 0.5 and increasing the lattice constant, the absorption edge moves to lower energies covering that way most of the visible spectrum (Fig. 4). In particular, the absorption is above 50% for energies higher than 2.6, 2, and 1.6 eV for a = 100, 200, 400 nm. For a = 400 nm (dotted line in Fig. 4), there is [(Fig._2)TD$IG]significant absorption above 10% even for energies
Fig. 2. The absorption of an array of Si nanowires with a = 300 nm and r/a = 0.25 (dash), 0.5 (dotted), 0.75 (dash-dotted line). The horizontal axis is the energy in eV of the incident light. The solid line corresponds to uniform Si slab of 4 mm thickness.
[(Fig._3)TD$IG]
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as low as 1.25 eV (in the IR region of the spectrum). There are however significant fluctuations due to the increased scattering of light from the arrays that causes the absorption to drop in some energies (e.g. at 2 eV the absorption is 0.22). The significant improvement of the absorption by increasing the lattice constant well above 100 nm has been also confirmed in another recent study [13]. [(Fig._5)TD$IG]
In the calculations so far a periodic arrangement of Si nanowires was considered. However, the fabricated arrays are made with chemical vapor deposition techniques and they are inherently random [3,7,9]. In particular, the scanning electron micrographs of the fabricated structures [3,7,9] show that the fabricated nanowires have disorder in radius, location and their orientation. It is therefore important to consider the effect of different types of disorder on their absorption spectrum. First, the effect of the disorder in radius is considered. In the calculations, a two by two supercell is considered containing four nanowires with separation a = 300 nm and average r/a = 0.5. Using a random number generator, the radius of each nanowire is changed with maximum percentage change from their average value equal to D. Finally, the absorption is calculated by averaging the results over 10 different configurations for each value of D. Comparing the results from Fig. 5, it is clear that the disorder increases the absorption from most of the energies. In addition it moves the absorption band to lower photon energies. In particular, the absorption is above 50% for energies higher that 1.7, 1.6, and 1.5 eV for D = 0, 20, and 40%, respectively. This can be explained by noticing that the mean filling ratio of the nanowires is actually increases using the above procedure and according to the results from Figs. 2 and 3 that should cause the absorption band to move to lower photon energies. The effect of randomness in the location of nanowires does not affect the absorption spectrum as much as the disorder in radius. Fig. 6 shows the absorption for a case where the separation between the nanowires is a = 300 nm and r/a = 0.5. A two by two supercell is used and the location of the four nanowires is randomly changed in each configuration from its average location with a maximum D1% change relative to a. An average over 10 different configurations is used for the calculation of the absorption. Comparing the
Fig. 5. The absorption spectrum of an array of Si nanowires with r/ a = 0.5 and a = 300 nm. The disorder in the radius is 0 (dotted), 20 (dash), and 40% (solid line).
Fig. 6. The absorption spectrum of an array of Si nanowires with r/ a = 0.5 and a = 300 nm. The disorder in the location of the nanowires is 0 (dotted), 20 (solid), and 40% (dash line).
Fig. 3. The absorption of an array of Si nanowires with a = 600 nm and r/a = 0.25 (solid), 0.5 (dash), 0.75 (dotted line).
[(Fig._4)TD$IG]
Fig. 4. The absorption of an array of Si nanowires with r/a = 0.5 and a = 100 (solid), 200 (dash), 400 nm (dotted line).
[(Fig._6)TD$IG]
[(Fig._7)TD$IG] 166
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Fig. 7. The absorption spectrum of an array of Si nanowires with r/ a = 0.5 and a = 300 nm. The disorder in the orientation of the nanowires is 0 (dash), p/10 (solid), and p/15 (dotted line).
disorder (solid and dash lines in Fig. 6) with the periodic case (dotted line in Fig. 6), one can conclude that there is small change for photon energies below 1.8 eV and a small increase (less than 20% in most energies) of the absorption for higher energies when the disorder is introduced. The disorder in the orientation is also studied. In that case the location and radius of the nanowires is a = 300 nm and r = 0.5a. Assuming that the axis of the unperturbed nanowires is z, the maximum change in the u angle is D2 while the f angle is randomly changed within the range [ p, p]. As in the previous random cases, an average over 10 different configurations is used for the calculation of the absorption. For D2 = p/15 (dotted line in Fig. 7), the results are very similar with the unperturbed case. However, increasing the disorder in the orientation (D2 = p/10, solid line in Fig. 7), the absorption in most photon energies decreases, indicating that this type of disorder may lead to reduction of the absorption of the nanowire arrays. To make the nanowire arrays more practical for solar cells applications, one has to place electrodes on the top and bottom surface of the arrays. One way to do that is by placing transparent electrodes (e.g. Indium Tin Oxide or ITO). Putting ITO on both sides of the nanowire arrays does not affect the absorption spectra [(Fig._8)TD$IG]significantly. However, placing a metal electrode on the
Fig. 8. The absorption spectrum of an array of Si nanowires with r/ a = 0.5 and a = 100 (solid), 300 (dash), 400 nm (dotted line). There is a thick layer of Ag at the bottom.
bottom has a significant effect. Fig. 8 shows the absorption spectrum of nanowire arrays having a thick (semi-infinite long in the calculations) homogeneous Ag layer at the bottom with r/a = 0.5 and different separations, a. The thickness of the nanowires is t = 4 mm as in the previous calculations. Comparing these results with the no Ag layer cases (see Fig. 4), one can see that the absorption edge moves to lower energies. For the Ag layer cases with a = 100 and 300 nm (Fig. 8), the absorption is higher than 50% for energies higher than 2.05 and 1.4 eV, respectively. Compare these values with the no Ag layer cases where the absorption is higher than 50% for energies higher than 2.6 and 1.6 eV, for a = 100 and 300 nm, respectively (see Figs. 2 and 4). Even for Ag layer case with a = 400 nm (dotted line in Fig. 8), the absorption increases for most of the energies relative to the no Ag layer case. In fact, the results of Fig. 8 are closer to the results of a similar nanowire system without the bottom Ag layer but with thickness of 8 mm. Besides some differences like the lower energy absorption bands for the case with Ag layer (Fig. 8) and most significant the absorption peak at 2.1 eV for the a = 100 nm (solid line in Fig. 8), the similarity of the results in Fig. 8 with the bare nanowires of thickness 8 mm show that the main effect of the uniform Ag layer is to reflect back the light doubling that way the effective length of the nanowires. A transparent electrode has to be placed on the top surface, so the sunlight can be transmitted to the nanowire arrays without significant reflections or absorption from this layer. Fig. 9 shows the absorption spectrum for the nanowire arrays with the thick bottom Ag layer and a thick layer (semi-infinite long) of ITO on the top. The refractive index of the ITO is assumed to be 1.4 and is independent of the photon energy. Comparing these results with the ones in Fig. 8 (no ITO layer), one can indeed see no significant changes confirming our expectations that ITO is ideal for top electrode.
[(Fig._9)TD$IG]
Fig. 9. The absorption spectrum of an array of Si nanowires with r/ a = 0.5 and a = 100 (solid), 300 (dash), 400 nm (dotted line). There is a thick layer of Ag at the bottom and a thick ITO layer on the top.
[(Fig._10)TD$IG]
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Fig. 10. The absorption spectrum of an array of nanowires with r/ a = 0.5 and a = 300 nm. The nanowires are made of InAs (solid), Ge (dash) and GaAs (dotted line).
Finally, although most of the work in nanowire solar cells arrays has been done with silicon, other materials may be also used. Fig. 10 shows the absorption spectrum for InAs, Ge and GaAs nanowires with r/ a = 0.5 and a = 300 nm, t = 4 mm and without the Ag and the ITO layer on bottom and the top of nanowires. Comparing with the corresponding results for Si (dash line in Fig. 2), one can see a significant increase of the absorption and a shift of the absorption edge toward the IR region for all of these materials. More particular, absorption above 50% appears for photon energies higher than 0.8, 1, 1.1 eV for InAs, Ge, and GaAs nanowires and that compares with 1.6 eV for Si nanowires. Although the absorption spectrum is very encouraging for all three materials, one has to keep in mind that there are other factors that are important in order for those materials to have applications in solar cells such as mobilities of the carriers, easy and cost of manufacturing them. 3. Conclusions The absorption of nanowire arrays was studied with the Rigorous Coupled Wave Analysis. The main goal of this study was to find the optimum conditions in order to use these systems for solar cell applications so they should have high absorption in as much as possible of the visible spectrum. Previous calculations of similar nanowire systems have studied the effects of the radius, separation and thickness of nanowires. The present work considered also the effects of different types of disorder, the effects of top and bottom electrodes (necessary for practical working devices), and the possibility of using other types of materials besides the commonly used Si. Square arrays of nanowire with separation around 300 nm and ratios of radius over separation between 0.5 and 0.75 are the most promising cases. The effect of the
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disorder, inevitable to appear in any fabricated devices, was also studied. The disorder in the radius increases the absorption while the disorder in the location of nanowires did not affect the absorption spectrum as much. In contrary, the disorder in the orientation of nanowires may reduce the absorption relative to corresponding unperturbed case. Adding a Ag layer as a bottom electrode and an ITO layer as a top electrode could increase the absorption in most of the visible spectrum. Although most of the studies are made with silicon nanowires, the calculations of the absorption spectrum showed that other materials, such as InAs, Ge, GaAs, may actually give higher absorption and they should certainly considered in future nanowire systems for solar cells applications. References [1] M.A. Green, Thin film solar cells: review of materials, technologies and commercial status, J. Mater. Sci.: Mater. Electron. 18 (2007) S15–S19. [2] B. Tian, T.J. Kempa, C.M. Lieber, Single nanowire photovoltaics, Chem. Soc. Rev. 38 (2009) 16–24. [3] L. Tsakalakos, J. Balch, J. Fronheiser, B.A. Korevaar, O. Suliman, J. Rand, Silicon nanowire solar cells, Appl. Phys. Lett. 91 (2007) 233117. [4] B. Tian, X. Zheng, T.J. Kempa, Y. Fang, N. Yu, G. Yu, J. Huang, C.M. Lieber, Coaxial silicon nanowires as solar cells and nanoelectronic power sources, Nature 449 (2007) 885–890. [5] M.D. Kelzenberg, D.B. Turner-Evans, B.M. Kayes, M.A. Filler, M.C. Putnam, N.S. Lewis, H.A. Atwater, Photovoltaic measurements in single nanowire silicon solar cells, Nano Lett. 8 (2008) 710–714. [6] L. Hu, G. Chen, Analysis of optical absorption in silicon nanowire arrays for photovoltaic applications, Nano Lett. 7 (2007) 3249–3252. [7] O.L. Muskens, J. Gomez Rivas, R.E. Algra, E.P.A.M. Bakkers, A. Lagendijk, Design of light scattering in nanowire materials for photovoltaic applications, Nano Lett. 8 (2008) 2638–2642. [8] J. Zhu, Z. Yu, G.F. Burkhard, C.-M. Hsu, S.T. Connor, Y. Xu, Q. Wang, M. McGehee, S. Fan, Y. Cui, Optical absorption enhancement in amorphous silicon nanowire and nanocone arrays, Nano Lett. 9 (2009) 279–282. [9] T. Stelzner, M. Pietsch, G. Andra, F. Falk, E. Ose, S. Christiansen, Silicon nanowire based solar cells, Nanotechnology 19 (2008) 295203. [10] M.G. Moharam, T.K. Gaylord, Rigorous coupled wave analysis of metallic surface relief gratings, J. Opt. Soc. Am. 3 (1986) 1780–1787. [11] L. Li, New formulation of the Fourier modal method for crossed surface relief gratings, J. Opt. Soc. Am. 14 (1997) 2758–2767. [12] E.D. Palik, Handbook of Optical Constants of Solids, Academic Press, London, UK, 1998. [13] C. Lin, M.L. Provinelli, Optical absorption enhancement in silicon nanowire arrays with large lattice constant for photovoltaic applications, Opt. Express 17 (2009) 19371.
Photonics and Nanostructures – Fundamentals and Applications 9 (2011) 163–167 www.elsevier.com/locate/photonics
The optical absorption of nanowire arrays N. Lagos a,b, M.M. Sigalas a,*, D. Niarchos b a
Institute of Materials Science, N.C.S.R ‘‘Demokritos’’ Agia Paraskevi, 15310 Athens, Greece b Materials Science Department, University of Patras, 26504 Patras, Greece
Received 25 January 2010; received in revised form 16 July 2010; accepted 2 September 2010 Available online 15 September 2010
Abstract The optical absorption of nanowire arrays is calculated with the rigorous coupled wave analysis. The effect of different parameters such as periodicity, filling ratio and thickness are studied. There are potential applications of these arrays for photovoltaic applications and there are experiments on Si nanowire arrays. For that reason the main objective of this study is to find the maximum absorption of nanowire arrays over a frequency area that covers the visible spectrum. There is a random location and orientation of the nanowires in those experiments. For that reason the effect of disorder in the absorption results is also examined. Although, the main focus of this study is Si nanowires, other materials are also calculated. # 2010 Elsevier B.V. All rights reserved. Keywords: Nanowires; Solar cells; Absorption; Disorder
1. Introduction There is a growing interest in photovoltaics (PV) since there is one of the main renewable energy sources. PVs based on thick (100 mm or thicker) silicon materials are still the main commercially used technology. One of the problems with this technology is the high amount of material used which accounts for almost half of the cost of those devices [1]. For that reason, there are intense efforts looking for thin films technologies as well as different materials that can give similar or even higher efficiencies with less material, therefore reduced costs [1]. One of the new directions in PVs is related with nanowire arrays [2–9]. Due to the bottom-up design followed in their fabrication, they can give good control of important parameters that affect the PV performance such as chemical and dopant composition, size and structure of
* Corresponding author. E-mail address: [email protected] (M.M. Sigalas). 1569-4410/$ – see front matter # 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.photonics.2010.09.005
the diode junction [2]. There are two types of nanowires, the axial where the p–i–n junction is formed along their axis and the radial where the p–i–n junction is formed along their radius. The radial nanowires have carrier collection distance smaller or comparable with the minority carrier diffusion length. That reduces significantly losses of the carriers in the bulk and increases the possibility of the photogenerated carriers to reach the contacts [2]. Here, the different parameters that affect the optical absorption are examined. The Rigorous Coupled Wave Analysis (RCWA) is used to calculate the reflection (R) and transmission (T) of electromagnetic waves from nanowire arrays [10,11]. The main goal of this structure is to find the optimum parameters (periodicity, filling ratio, material) that give the maximum absorption (A = 1 T R) for the maximum possible visible frequency area. The effect of the disorder is also studied since in all the fabricated nanowire arrays is always present. The dielectric constants used in the calculations are from measurements [12]. In particular, both Si as
[(Fig._1)TD$IG] 164
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Fig. 1. The xz and xy cross-sections of the nanowire structure.
well as the other dielectric materials studied and Ag have frequency dependent dielectric constants that are taken by fitting the measured results [12]. It should be mentioned that RCWA is a frequency domain method and it needs the dielectric constant in each frequency. 2. Results and discussion The structure used in the calculations is shown in Fig. 1. It is a square lattice with lattice constant, a, and radius of the nanowires, r. The incident light is along the z-axis and polarized along the x-axis. The calculations show that increasing the thickness of the nanowires their absorption also increases in accordance with previous studies [6]. They also show that there are no significant changes in the absorption spectrum for incident angles less that 208 for both polarizations. So, all the following results are for zero incident angle and t = 4 mm thick nanowires. For lattice constant a = 300 nm, the absorption as a function of energy is shown in Fig. 2. The general trend of the absorption resembles a step function. There is high absorption at high energies where Si is highly absorbing and lower absorption at lower energies. The goal of the present study is to find the optimum parameters for which the high absorption area can cover as much as possible of the visible spectrum and if possible some part of the near IR region since these are the areas where most of the sunlight energy is concentrated. One way to achieve that is to increase the radius of the nanowires. For r/a = 0.25, the absorption is almost 100% for energies higher than 2.9 eVand drops below 10% for energies less than 2.5 eV (Fig. 2). Increasing the radius to 0.5a, the absorption stays above 50% for energies above 1.6 eV covering all the visible spectrum. The drop of the absorption at around 2.4 eV is due to the increased scattering from the nanowire arrays that leads to higher reflection. Increasing the radius to 0.75a, the absorption is between 70% and 90% for energies above 2.2 eV and drops below 50% for
energies higher than 1.5 eV similarly as in the r/a = 0.5 case. As a comparison, Fig. 2 also shows the uniform Si case with thickness of 4 mm, where the maximum absorption is 64% at 2.2 eV. The absorption is higher than the uniform Si slab, for most of the visible spectrum for the cases with r/a = 0.5 and 0.75. Increasing the lattice constant, the scattering of the light from the array becomes more significant since the wavelength of light becomes comparable with the feature sizes. That becomes clear for a = 600 nm (Fig. 3), where the absorption spectrum for all radius show significant fluctuations. For r/a = 0.25, the absorption is above 20% but lower than 50%, for all the energies above 1.8 eV. Increasing the radius to 0.5a, increases the absorption at higher energies and significant absorption above 10% appears for energies as low as 0.9 eV. However, the r/a = 0.75 has the best absorption spectrum since it remains above 70% for most of the energies above 1.6 eV covering the whole visible spectrum. Keeping the r/a equal to 0.5 and increasing the lattice constant, the absorption edge moves to lower energies covering that way most of the visible spectrum (Fig. 4). In particular, the absorption is above 50% for energies higher than 2.6, 2, and 1.6 eV for a = 100, 200, 400 nm. For a = 400 nm (dotted line in Fig. 4), there is [(Fig._2)TD$IG]significant absorption above 10% even for energies
Fig. 2. The absorption of an array of Si nanowires with a = 300 nm and r/a = 0.25 (dash), 0.5 (dotted), 0.75 (dash-dotted line). The horizontal axis is the energy in eV of the incident light. The solid line corresponds to uniform Si slab of 4 mm thickness.
[(Fig._3)TD$IG]
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as low as 1.25 eV (in the IR region of the spectrum). There are however significant fluctuations due to the increased scattering of light from the arrays that causes the absorption to drop in some energies (e.g. at 2 eV the absorption is 0.22). The significant improvement of the absorption by increasing the lattice constant well above 100 nm has been also confirmed in another recent study [13]. [(Fig._5)TD$IG]
In the calculations so far a periodic arrangement of Si nanowires was considered. However, the fabricated arrays are made with chemical vapor deposition techniques and they are inherently random [3,7,9]. In particular, the scanning electron micrographs of the fabricated structures [3,7,9] show that the fabricated nanowires have disorder in radius, location and their orientation. It is therefore important to consider the effect of different types of disorder on their absorption spectrum. First, the effect of the disorder in radius is considered. In the calculations, a two by two supercell is considered containing four nanowires with separation a = 300 nm and average r/a = 0.5. Using a random number generator, the radius of each nanowire is changed with maximum percentage change from their average value equal to D. Finally, the absorption is calculated by averaging the results over 10 different configurations for each value of D. Comparing the results from Fig. 5, it is clear that the disorder increases the absorption from most of the energies. In addition it moves the absorption band to lower photon energies. In particular, the absorption is above 50% for energies higher that 1.7, 1.6, and 1.5 eV for D = 0, 20, and 40%, respectively. This can be explained by noticing that the mean filling ratio of the nanowires is actually increases using the above procedure and according to the results from Figs. 2 and 3 that should cause the absorption band to move to lower photon energies. The effect of randomness in the location of nanowires does not affect the absorption spectrum as much as the disorder in radius. Fig. 6 shows the absorption for a case where the separation between the nanowires is a = 300 nm and r/a = 0.5. A two by two supercell is used and the location of the four nanowires is randomly changed in each configuration from its average location with a maximum D1% change relative to a. An average over 10 different configurations is used for the calculation of the absorption. Comparing the
Fig. 5. The absorption spectrum of an array of Si nanowires with r/ a = 0.5 and a = 300 nm. The disorder in the radius is 0 (dotted), 20 (dash), and 40% (solid line).
Fig. 6. The absorption spectrum of an array of Si nanowires with r/ a = 0.5 and a = 300 nm. The disorder in the location of the nanowires is 0 (dotted), 20 (solid), and 40% (dash line).
Fig. 3. The absorption of an array of Si nanowires with a = 600 nm and r/a = 0.25 (solid), 0.5 (dash), 0.75 (dotted line).
[(Fig._4)TD$IG]
Fig. 4. The absorption of an array of Si nanowires with r/a = 0.5 and a = 100 (solid), 200 (dash), 400 nm (dotted line).
[(Fig._6)TD$IG]
[(Fig._7)TD$IG] 166
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Fig. 7. The absorption spectrum of an array of Si nanowires with r/ a = 0.5 and a = 300 nm. The disorder in the orientation of the nanowires is 0 (dash), p/10 (solid), and p/15 (dotted line).
disorder (solid and dash lines in Fig. 6) with the periodic case (dotted line in Fig. 6), one can conclude that there is small change for photon energies below 1.8 eV and a small increase (less than 20% in most energies) of the absorption for higher energies when the disorder is introduced. The disorder in the orientation is also studied. In that case the location and radius of the nanowires is a = 300 nm and r = 0.5a. Assuming that the axis of the unperturbed nanowires is z, the maximum change in the u angle is D2 while the f angle is randomly changed within the range [ p, p]. As in the previous random cases, an average over 10 different configurations is used for the calculation of the absorption. For D2 = p/15 (dotted line in Fig. 7), the results are very similar with the unperturbed case. However, increasing the disorder in the orientation (D2 = p/10, solid line in Fig. 7), the absorption in most photon energies decreases, indicating that this type of disorder may lead to reduction of the absorption of the nanowire arrays. To make the nanowire arrays more practical for solar cells applications, one has to place electrodes on the top and bottom surface of the arrays. One way to do that is by placing transparent electrodes (e.g. Indium Tin Oxide or ITO). Putting ITO on both sides of the nanowire arrays does not affect the absorption spectra [(Fig._8)TD$IG]significantly. However, placing a metal electrode on the
Fig. 8. The absorption spectrum of an array of Si nanowires with r/ a = 0.5 and a = 100 (solid), 300 (dash), 400 nm (dotted line). There is a thick layer of Ag at the bottom.
bottom has a significant effect. Fig. 8 shows the absorption spectrum of nanowire arrays having a thick (semi-infinite long in the calculations) homogeneous Ag layer at the bottom with r/a = 0.5 and different separations, a. The thickness of the nanowires is t = 4 mm as in the previous calculations. Comparing these results with the no Ag layer cases (see Fig. 4), one can see that the absorption edge moves to lower energies. For the Ag layer cases with a = 100 and 300 nm (Fig. 8), the absorption is higher than 50% for energies higher than 2.05 and 1.4 eV, respectively. Compare these values with the no Ag layer cases where the absorption is higher than 50% for energies higher than 2.6 and 1.6 eV, for a = 100 and 300 nm, respectively (see Figs. 2 and 4). Even for Ag layer case with a = 400 nm (dotted line in Fig. 8), the absorption increases for most of the energies relative to the no Ag layer case. In fact, the results of Fig. 8 are closer to the results of a similar nanowire system without the bottom Ag layer but with thickness of 8 mm. Besides some differences like the lower energy absorption bands for the case with Ag layer (Fig. 8) and most significant the absorption peak at 2.1 eV for the a = 100 nm (solid line in Fig. 8), the similarity of the results in Fig. 8 with the bare nanowires of thickness 8 mm show that the main effect of the uniform Ag layer is to reflect back the light doubling that way the effective length of the nanowires. A transparent electrode has to be placed on the top surface, so the sunlight can be transmitted to the nanowire arrays without significant reflections or absorption from this layer. Fig. 9 shows the absorption spectrum for the nanowire arrays with the thick bottom Ag layer and a thick layer (semi-infinite long) of ITO on the top. The refractive index of the ITO is assumed to be 1.4 and is independent of the photon energy. Comparing these results with the ones in Fig. 8 (no ITO layer), one can indeed see no significant changes confirming our expectations that ITO is ideal for top electrode.
[(Fig._9)TD$IG]
Fig. 9. The absorption spectrum of an array of Si nanowires with r/ a = 0.5 and a = 100 (solid), 300 (dash), 400 nm (dotted line). There is a thick layer of Ag at the bottom and a thick ITO layer on the top.
[(Fig._10)TD$IG]
N. Lagos et al. / Photonics and Nanostructures – Fundamentals and Applications 9 (2011) 163–167
Fig. 10. The absorption spectrum of an array of nanowires with r/ a = 0.5 and a = 300 nm. The nanowires are made of InAs (solid), Ge (dash) and GaAs (dotted line).
Finally, although most of the work in nanowire solar cells arrays has been done with silicon, other materials may be also used. Fig. 10 shows the absorption spectrum for InAs, Ge and GaAs nanowires with r/ a = 0.5 and a = 300 nm, t = 4 mm and without the Ag and the ITO layer on bottom and the top of nanowires. Comparing with the corresponding results for Si (dash line in Fig. 2), one can see a significant increase of the absorption and a shift of the absorption edge toward the IR region for all of these materials. More particular, absorption above 50% appears for photon energies higher than 0.8, 1, 1.1 eV for InAs, Ge, and GaAs nanowires and that compares with 1.6 eV for Si nanowires. Although the absorption spectrum is very encouraging for all three materials, one has to keep in mind that there are other factors that are important in order for those materials to have applications in solar cells such as mobilities of the carriers, easy and cost of manufacturing them. 3. Conclusions The absorption of nanowire arrays was studied with the Rigorous Coupled Wave Analysis. The main goal of this study was to find the optimum conditions in order to use these systems for solar cell applications so they should have high absorption in as much as possible of the visible spectrum. Previous calculations of similar nanowire systems have studied the effects of the radius, separation and thickness of nanowires. The present work considered also the effects of different types of disorder, the effects of top and bottom electrodes (necessary for practical working devices), and the possibility of using other types of materials besides the commonly used Si. Square arrays of nanowire with separation around 300 nm and ratios of radius over separation between 0.5 and 0.75 are the most promising cases. The effect of the
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disorder, inevitable to appear in any fabricated devices, was also studied. The disorder in the radius increases the absorption while the disorder in the location of nanowires did not affect the absorption spectrum as much. In contrary, the disorder in the orientation of nanowires may reduce the absorption relative to corresponding unperturbed case. Adding a Ag layer as a bottom electrode and an ITO layer as a top electrode could increase the absorption in most of the visible spectrum. Although most of the studies are made with silicon nanowires, the calculations of the absorption spectrum showed that other materials, such as InAs, Ge, GaAs, may actually give higher absorption and they should certainly considered in future nanowire systems for solar cells applications. References [1] M.A. Green, Thin film solar cells: review of materials, technologies and commercial status, J. Mater. Sci.: Mater. Electron. 18 (2007) S15–S19. [2] B. Tian, T.J. Kempa, C.M. Lieber, Single nanowire photovoltaics, Chem. Soc. Rev. 38 (2009) 16–24. [3] L. Tsakalakos, J. Balch, J. Fronheiser, B.A. Korevaar, O. Suliman, J. Rand, Silicon nanowire solar cells, Appl. Phys. Lett. 91 (2007) 233117. [4] B. Tian, X. Zheng, T.J. Kempa, Y. Fang, N. Yu, G. Yu, J. Huang, C.M. Lieber, Coaxial silicon nanowires as solar cells and nanoelectronic power sources, Nature 449 (2007) 885–890. [5] M.D. Kelzenberg, D.B. Turner-Evans, B.M. Kayes, M.A. Filler, M.C. Putnam, N.S. Lewis, H.A. Atwater, Photovoltaic measurements in single nanowire silicon solar cells, Nano Lett. 8 (2008) 710–714. [6] L. Hu, G. Chen, Analysis of optical absorption in silicon nanowire arrays for photovoltaic applications, Nano Lett. 7 (2007) 3249–3252. [7] O.L. Muskens, J. Gomez Rivas, R.E. Algra, E.P.A.M. Bakkers, A. Lagendijk, Design of light scattering in nanowire materials for photovoltaic applications, Nano Lett. 8 (2008) 2638–2642. [8] J. Zhu, Z. Yu, G.F. Burkhard, C.-M. Hsu, S.T. Connor, Y. Xu, Q. Wang, M. McGehee, S. Fan, Y. Cui, Optical absorption enhancement in amorphous silicon nanowire and nanocone arrays, Nano Lett. 9 (2009) 279–282. [9] T. Stelzner, M. Pietsch, G. Andra, F. Falk, E. Ose, S. Christiansen, Silicon nanowire based solar cells, Nanotechnology 19 (2008) 295203. [10] M.G. Moharam, T.K. Gaylord, Rigorous coupled wave analysis of metallic surface relief gratings, J. Opt. Soc. Am. 3 (1986) 1780–1787. [11] L. Li, New formulation of the Fourier modal method for crossed surface relief gratings, J. Opt. Soc. Am. 14 (1997) 2758–2767. [12] E.D. Palik, Handbook of Optical Constants of Solids, Academic Press, London, UK, 1998. [13] C. Lin, M.L. Provinelli, Optical absorption enhancement in silicon nanowire arrays with large lattice constant for photovoltaic applications, Opt. Express 17 (2009) 19371.