A hybrid approach for light trapping using photonic crystal nanostructures (nanorods, nanopillars or nanoholes) on top of an ultra thin film as a substrate is presented. The combination of a nanopatterned layer with a thin substrate shows an enhanced optical absorption than equivalent films without patterning and can compete in performance with nanostructured systems without a substrate. The designs are tested in four relevant materials: amorphous silicon (a-Si), crystalline silicon (Si), gallium arsenide (GaAs) and indium phosphide (InP). A consistent enhancement is observed for all of the materials when using a thin hybrid system (300 nm) even compared to the non patterned thin film with an anti-reflective coating (ARC). A realistic solar cell structure composed of a hybrid system with a layer of indium tin oxide (ITO) an ARC and a back metal layer is performed, showing an 18% of improvement for the nanostructured device.
© 2012 OSA
Solar cells based on thin or ultra thin films have potential advantages like a reduction in the amount of absorbing material and a better extraction of the photogenerated carriers. Light trapping techniques may help to reduce the total thickness of the layers involved without losing efficiency in the absorption of light. Periodic nanostructured surfaces like photonic crystals have been demonstrated to improve optical absorption and consequently ultimate efficiency [1–5]. Some of those works present simulations of completely patterned or nanostructured systems, like nanoholes (NHoles) or nanopillars (NPillars) without a substrate [1, 4, 5]. In contrast we have focused our work on the use of a nanostructured layer (either NHoles or NPillars) on top of a thin substrate. This hybrid approach presents several advantages: 1) from a technological point of view, the substrate provides a more robust mechanical support for the nanostructured layer 2) the etched thickness is reduced compared to a totally nanostructured layer, i.e. without a substrate 3) the substrate helps to improve the absorption of light coupled to it by the photonic crystal layer 4) there is an anti-reflection effect due to the nanopatterning [6, 7] and 5) the modes inside the photonic crystal are also hybrid (not pure TE or TM) due to the broken symmetry [8, 9] increasing the total number of available modes of incident light . During the revision of the present work we became aware of the publication of  where a periodic system on top of a substrate is also used.
The proposed photonic crystals (PCs) are composed of nanorods (NRods) for a one dimensional (1D) case and NPillars or NHoles for two dimensional (2D) systems (Fig. 1 ). The total thickness of the nanostructured and substrate layers has been always kept to less than 1 µm. The change in absorption has been studied with the variation of the total thickness of the hybrid system. Normal and oblique incidence have been also tested. The ultimate efficiencies of nanopatterned systems have been compared to thin film (TF) with and without a single ARC layer. Finally, a realistic case of a solar cell device composed of an absorbing nanostructured layer, an ARC, an ITO electrode and a back metal layer has been compared to the non patterned case.
For the photonic crystal layer, a square symmetry has been chosen with the same lattice parameter for all the structures, a = 450 nm, which provides a good value when the useful part of the solar spectrum is taken in account [1, 2]. The width of the NRods (D) and the diameter of the NPillars and NHoles (d) are 225 nm in both cases, being a reasonable compromise between performance and current techniques of fabrication. The nanopatterning of the surface means less amount of absorbing material than in the case of a non-structured system, which should be taken into account to compare the different systems. In the present work we have not focus our interest in obtaining an optimized design, but instead on the exploration of the hybrid photonic crystal-substrate system for relevant photovoltaic materials (Si, a-Si, GaAs, InP) with a realistic geometry and with the addition of an ARC layer.
2. Theory and numerical methods
We have calculated the absorption spectra from an incident flux of planar waves with energies ranging from 1 eV to 4 eV (1240 nm to 310 nm). Reflection (R) and transmission (T) is simulated and the absorption (A) is obtained as A = 1-T-R. For the different structures and materials we have calculated the ultimate efficiency  defined as:13]. For a perfect absorber (PA),, the ultimate efficiencies are for GaAs, for InP, for Si and for a-Si. We define the PC filling factor in 1D systems as , and for 2D systems as . The corresponding filling factors were, and for for NRods, NHoles and NPillars respectively. The dielectric functions of the absorbing materials have been taken from . We have modeled the optical properties of the nanostructured solar cells using a three-dimensional finite difference time domain (3D-FDTD) method . A grid of 5 nm was chosen, and 100 frequencies from 1 to 4 eV were calculated. For a faster calculation we have fitted the complex dielectric permittivity of the material to a Lorentz model:14]. In Fig. 2 the fits are shown along the data taken from .
3.1 1D systems
We have studied 1D systems because they may be easier to fabricate than 2D structures and may offer a similar performance. For 1D systems it is necessary to calculate the light incident with s and p polarizations [1,2]. We have tested our structures using an unpolarized source, mixing the two polarizations with a randomly time dependent phase between them. The 1D structure is composed of in-plane contained NRods with a height h = 150 nm over a thin film substrate of the same material and height. Figures 3 and 4 show the absorption and reflection calculated for the 1D systems using GaAs, InP, Si, a-Si and a 300nm-thick TF for comparison.
3.2 2D systems
2D photonic systems without any substrate have been tested in [4, 5]. The thicknesses in both cases were about ten times higher than in this work. Those systems have been shown to present a better coupling with light than non patterned structures . As in the 1D case, our 2D system is composed of NHoles or NPillars with a thickness h = 150 nm which are on top of a substrate with the same thickness and material than the nanostructured layer. Figures 5 and 6 show the absorption and reflection calculated for the 2D systems and for the 300nm-thick TF.
Table 1 shows the ultimate efficiencies obtained for the nanopatterned systems described above and for the different materials used. The absorption of the non-patterned TF with a thickness of 300 nm is shown for comparison. We have included the case of the non-patterned TF with a single layer ARC optimized in thickness between 0 and 300 nm to maximize the ultimate efficiency. For the silicon-based systems the ARC material was SiO2 (80 nm for Si and 80 nm for a-Si) and for the systems based on III-V semiconductors we used Si3N4 (60 nm for GaAs and 70 nm for InP). As a figure of comparison we define the increment in the efficiencies normalized to the PA for the case with and without an ARC, respectively, as: , where is the efficiency of the nanopatterned material, is the efficiency of the non-patterned structure with the same amount of absorbing material, is the efficiency with the ARC, and is the ultimate efficiency for the perfect absorber. The values for the figures of comparison and are given for the system with the highest enhancement (i.e. ). From the obtained values, it is remarkable that a persistent enhancement for all of the nanopatterned systems is present despite they do not include any ARC layer.
From Table 1 we can conclude that, at least for the simulated materials, the systems with a higher enhancement in the absorption use NPillars, although the difference with the NRod (1D-case) is not very high. Despite the NHoles do not show a higher enhancement than NPillars for the direct bandgap materials, for crystalline Si the calculated enhancement is equal for both. In order to explore the effect of the thickness in our 2D systems we have evaluated the absorption of NPillars and NHoles systems for the following cases: i) 150 nm PC + 150 nm TF substrate ii) 300 nm PC + 300 nm TF substrate iii) 500 nm PC + 500 nm TF substrate. Thicknesses above one micron are close to the perfect absorption limit for III-V semiconductors, so we have kept the calculation below that value. The results are displayed in Fig. 7 and Fig. 8 .
When increasing equally the thickness of both the nanopatterned layer and the substrate, the NHoles do not show further improvement for high energies, whereas the NPillars do. Also, the difference between enhancements of the absorption for the two systems is not constant with the change in the thickness. This marks a different behavior between the two systems (NHoles and NPillars). The ultimate efficiencies for Si, GaAs and InP (a-Si is not considered for the short diffusion length of the minority carriers) are shown in Table 2 .
If we take the case of Si, the NHoles have better efficiency than NPillars for the ii) structure. This is related to the presence of guided modes inside the substrate and suggests that for an optimized design the total thickness of each of the layers has to be taken into account. The nanopatterned system is better for the thinnest systems, as it shows the i)-type structures. On the other side, for the case of thick layers of III-V semiconductors, the nanopatterned system is not necessarily better than a simple ARC layer, with increments of efficiency that can reach even negative values. This is expected since the GaAs and InP have direct bandgap absorbing almost all the photons in a thickness about one micron. Therefore, patterned and non-patterned systems show similar performance. The Si, with an indirect bandgap, is unable to absorb all the useful photons in one micron, even with an ARC which helps to couple light to the semiconductor. We attribute the achievement of the highest efficiencies for the thinnest systems, independently of the material, to the photonic resonances inside the photonic crystal layer, which are more pronounced as the confinement of the light increases, as it happens when total thickness decreases. This effect has been shown in crystalline Si and a-Si  and for III-V direct bandgap semiconductors in this work. Further work is needed to understand in depth this coupling.
In general, the structures based on NPillars seem to have higher efficiencies than the ones based in NHoles, so we have calculated their performance against light in non-normal incidence to clarify if this enhancement disappears with the angle. We have used the same parameters as above and GaAs as the absorbing material, which we expect to behave in a similar way than the rest of the materials with high absorption. We have set the p polarization parallel to the Y direction of the square symmetry of the pattern. The system is illuminated for angles of 0°(normal incidence), 15°, 30°, and 45°.
The results are displayed in Fig. 9 and Fig. 10 . The enhancement in the ultimate efficiency does not show a drastic decrease when changing the incidence. For example, at 45°the ultimate efficiency decays only to 0.30 whereas at normal incidence is 0.33. A similar behaviour has been obtained for NHoles with an ultimate efficiency of 0.27 for normal incidence, which decays to 0.25 at 45°.
3.3 Enhancement factor in the weak absorption limit
As it has been explained on , the enhancement factor limit is valid for light trapping in bulk solar cells with low intrinsic absorption and when using geometric optics. The last assumption is not satisfied by our systems because of the sub-wavelength size of the nanopatterning. Nevertheless, it may be useful to compare those systems with the Lambertian absorber (LA), as shown in Figs. 3, 5 and 7.
The absorption for the LA is higher than the nanopatterned systems for almost all energies except for the systems made of Si. Si is the only material in this work with an intrinsic weak absorption so the assumptions made for the LA make more sense. The other systems, GaAs, InP and a-Si are above the limits of , () so the absorption enhancement is supposed to be not related to multiple reflections inside the semiconductor. The enhancement factor for a normalized source () is calculated as, (). The silicon systems (NHoles and NPillars) between 1.12 and 1.5 eV (energy range for weak absorption) were calculated using Lumerical FDTD solutions simulation package, which shows a better numerical stability in this region.
Figure 11 show that the absorptions for all of the nanopatterned systems are above the Lambertian limit. We attribute that to a photonic light trapping regime. In a similar way this behavior has been described in , where thin film systems can overtake the Lambertian limit. As energies increase, the enhancement factor decreases, which is expected as the extinction coefficient increases and diverges from the weak absorption regime. As it has been previously described  another figure of comparison can be used by defining a ratio between the absorptions, Lambertian or Photonic, and the single pass absorption (). This has been calculated for the thinnest system (2h = 300 nm) and all of the materials (GaAs, InP, Si and a-Si) as Fig. 12 shows.
3.4 An a-Si solar cell with ARC and back metal contact
We have simulated a simple case of a device composed of an a-Si layer which is nanostructured in half of its thickness, and an ARC composed by a SiO2 layer that completely covers the nanopatterned surface, planarizing it. The optimized thickness of the ARC is obtained for 80 nm above the nanopattern, as shown in Fig. 13 .
We have finally added a back metal contact formed by a perfect metal. Figure 14 shows the absorption and reflection for the patterned and non-patterned device.
The reflection of the non-patterned solar cell shows a peak around 1.5 eV related to the finite thickness of the total structure. The optimized ARC is calculated for the best ultimate efficiency but is not able to minimize the modulation associated with the thickness for every wavelength. This modulation effect can be very important in ultra thin films. The nanostructured solar cell with the ARC does not present that feature. This means that the nanostructured pattern with a substrate is useful to selectively remove modulations of the reflection or at least to smooth them. We attribute the elimination of the modulation in reflection and absorption to the coupling of photonic crystal quasi-guided modes and the incident light. This effect has been observed before in photonic crystal surfaces [3, 7, 18].
The ultimate efficiency for the non-patterned device is whereas for both the NPillars and NHoles are . Both types of nanostructures show higher absorptions than the thin film, but surprisingly there is no difference using NHoles or NPillars, contrary to what is obtained without the ARC layer (Table 1). When normalized to the maximum efficiency for a perfect absorber, the ultimate efficiency () for the hybrid system is 85%, whereas for the thin film is 72%, which means an increment () of 13%.
3.5 An a-Si solar cell with ARC, ITO electrode and back metal contact
An approach to a more realistic model has been done including a transparent conductive oxide (ITO) layer with a complex refractive index taken from . The layer is used as electrode for collecting the carriers (Fig. 15 ). A typical thickness for ITO to collect efficiently the carriers is about 150-200 nm, which is not an optimal design as ARC. Therefore, a second layer of SiO2 is included as an ARC coating. A local optimization of the thickness has been made for the ITO and SiO2 layers, obtaining nm (range of optimization was 150 to 200 nm) and s´ = 180 nm for the SiO2 (range of optimization from 0 to 600 nm). The NHoles and NPillars are filled or immersed with SiO2, with the contact on top. The absorption was calculated excluding absorption from the ITO. Therefore, the ultimate efficiencies were calculated using only the absorption of the semiconductor layer. Results are displayed in Fig. 16 .
The non-patterned system presents a high oscillation in the reflection and absorption, due to the thin film, that does not appear in the nanostructured ones. This was also observed in the system with SiO2 and a metal and shows the capacity of the nanostructured layer to smooth such modulations. The ultimate efficiency for the non-patterned device is (63% normalized to the perfect absorber), for NPillars is (i.e. 80%) and for the NHoles is (i.e. 82%). This means a relative increment of 17% and 19%, respectively, for the hybrid systems.
We have calculated the ultimate efficiencies of thin hybrid systems composed of a photonic crystal (nanopillars and nanoholes in 2D and nanorods in 1D) on top of a substrate. We have demonstrated that the hybrid system that shows a highest enhancement is always the thinnest system (300 nm) even if an ARC is included. This happens for the four materials used (a-Si, crystalline Si, GaAs and InP). Above that thickness a divergence in the ultimate efficiency appears between materials with direct and indirect bandgap showing that direct bandgap systems with an ARC can be similar in efficiency as nanopatterned systems. In this case, a final optimization depending on the thickness of both substrate and nanopattern will be needed for each material. A realistic case of a solar cell with a-Si, an absorbing ITO layer, an ARC and a back metal contact shows an enhancement of 18% respect to the cell without patterning. Modulations of absorption and reflection in the non patterned thin film disappear when the photonic crystal is included. For the thinnest systems (300nm) the efficiencies of nanopillars are higher than nanoholes when direct gap materials are used. For Si and for the realistic case of the a-Si solar cell with (or without) ITO and an ARC layer, nanopillars and nanoholes show almost the same efficiencies.
We acknowledge financial support by MICINN (grants ENE2009-14481-C02-02, CSD2006-0004, Innpacto IPT-2011-1467- 420000), and CAM (S2009/ENE-1477).
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