We propose a new figure of merit to assess the performance of light trapping nanostructures for solar cells, which we call the light trapping efficiency (LTE). The LTE has a target value of unity to represent the performance of an ideal Lambertian scatterer, although this is not an absolute limit but rather a benchmark value. Since the LTE aims to assess the nanostructure itself, it is, in principle, independent of the material, fabrication method or technology used. We use the LTE to compare numerous proposals in the literature and to identify the most promising light trapping strategies. We find that different types of photonic structures allow approaching the Lambertian limit, which shows that the light trapping problem can be approached from multiple directions. The LTE of theoretical structures significantly exceeds that of experimental structures, which highlights the need for theoretical descriptions to be more comprehensive and to take all relevant electro-optic effects into account.
© 2014 Optical Society of America
It is now well established that solar cells can make an important contribution to the renewable energy mix, with more than 100 GWp of capacity already installed . The main impediment to further growth is the price/performance ratio; we need to reduce cost by using less material or increase efficiency by converting more of the incoming solar radiation into photocurrent, or, ideally, both. Using nanophotonic techniques, i.e. “light trapping”, is a promising strategy to achieving these goals, and to enable the realisation of thinner solar cells with higher efficiency. Thinner solar cells may also benefit from higher open-circuit voltages due to reduced impact of bulk recombination: for crystalline silicon, for example, the limiting value increases from ca. 750 mV for a 300 µm thick cell to 830 mV for a 1 µm thick cell .
A large variety of light trapping structures has already been proposed and demonstrated in the literature; in order to identify the most promising structures, it is clearly important to describe their performance objectively and quantitatively. Typically, authors compare their proposed structures to an unstructured thin film and to an ideal Lambertian scatterer, the latter being a theoretical model derived from statistical ray optics considerations . The Lambertian scatterer enhances the path length up to 4n2 on average, where n stands for the refractive index of the absorbing material. The problem with this comparison is that it depends on the thickness of the absorber material and on the quality of the material itself. Further, the 4n2 enhancement can be approached only in case of weak active absorption : in other cases the active absorption has to be calculated by considering the attenuation of the single angular components of the scattered photon flux [4–6]. In the solar cells community it is well known that the Lambertian limit can be easily overcome at single wavelengths using different types of photonic approaches. Indeed, the main problem is to achieve enhancement at all (or at least most) wavelengths: for this reason we limited our analysis to works that report data calculated over the standard AM 1.5G spectrum . In addition, authors often use different materials, substrates and different model assumptions to assess the performance of their respective structures. Experimental structures may also suffer from parasitic absorption in the oxide layers and in the electrodes, which theoretical models tend to ignore.
In order to provide a unified description of light trapping properties and to take this large variety of effects into account, we propose a new figure of merit to describe the light trapping performance, which we term the light trapping efficiency (LTE). The LTE aims at assessing the performance of the nanostructure itself, irrespective of the material, fabrication method and technology used.
2. The format of the LTE figure of merit
The LTE uses the short-circuit current JSC as a basis for describing the performance of a solar cell device; JSC measures the number of electron-hole pairs generated by the incoming solar flux, which is the parameter that light trapping is aiming to increase.
We calculate the JSC for the AM1.5 spectrum with the global irradiance of 100 mW/cm2 on Earth  and assume an internal quantum efficiency of unity, as outlined in more detail below (see Eqs. (1-4). Figure 1 shows the resulting values of JSC as a function of thickness for three extreme cases; a) JLL, for “Lambertian Limit”, assuming an ideal Lambertian scatterer with no external reflection losses (perfect anti-reflection coating) on the front surface and a perfect metal back-reflector, which yields the top (red dashed) curve; b) JMB, for “Metal Back-reflector”, which assumes an unstructured, perfectly planar thin film with perfect anti-reflection coating and metal back-reflector, represented by the blue solid line, and c) Jmin, the same as b) but without anti-reflection coating, which yields the bottom (black dotted) curve.
The JLL of an ideal solar cell assumes an internal quantum efficiency of unity for all wavelengths λ, maximum pathlength enhancement due to scattering (see appendix 2), a perfect anti-reflection coating and a perfect metal back-reflector:
JLL is always lower than the incident solar flux Jsun expressed as an electrical current of ~46 mA/cm2 for the specified wavelength interval of the global AM1.5 solar spectrum dIsun/dλ. The physical constants e, h and c are the electron charge, Planck’s constant and the velocity of light, respectively. In the case that light is fully randomized in a slab with thickness ttot, its transmittance Tr can be expressed by an angle-averaged effective absorption coefficient αeff [5,8,9]:
Tr is always smaller or equal to the transmittance Tsp of a non-randomized single pass traversal, because for weakly absorbed light (αttot << 1) the enhancement factor describing randomisation alone, which can be expressed as (1-Tr)/(1-Tsp), is at best 2, while it is unity for strongly absorbed light (αttot >> 1). Therefore, Tr measures Lambertian transmission through an absorbing layer and stands in contrast to the Lambertionality factor a from Battaglia et al. , which is independent of the material constant α.
The JMB corresponds to the short-circuit current generated by a double pass traversal of light in an ideal unstructured reference device, i.e. a solar cell with internal quantum efficiency equal to unity for all wavelengths λ, with a perfect anti-reflection coating and a perfect metal back-reflector:
The difference JLL – JMB can then be understood as the maximum theoretical current gain due to light trapping. In addition, avoiding reflections is another major issue in solar cells and photonic patterns. To highlight the importance of anti-reflection, we consider the case of a slab without any coating, for which the short-circuit current Jmin is calculated as follows:
When looking at literature proposals (c.f. section 3), Jmax denotes the current of a proposed light trapping design, while we name with Jref the short circuit current of its (unpatterned) reference device. Real solar cells may not achieve the theoretical values JLL or JMB, respectively, because of material imperfections, parasitic absorption or imperfect anti-reflection coating. The term Jmax – Jref therefore represents the improvement in current achieved by the real structure.
The LTE then compares the total current gain Jmax – Jref achieved by the real structure to the theoretical maximum current gain of the ideal Lambertian scatterer JLL – JMB, so our expression for the light trapping efficiency (LTE) takes the format
We make the following comments and assumptions:
- Light trapping for photovoltaic applications aims to increase the absorption over the full wavelength range of the global solar spectrum (from 300 nm up to the wavelength bandgap). All currents therefore refer to the full AM1.5G standard spectrum .
- All practical solar cells use a back-reflector for doubling the optical path length and for increasing the absorption probability of light. Therefore, JLL and JMB represent ideal solar cells with mirrors that exhibit no parasitic absorption, while Jmax and Jref include the properties of real mirrors.
- Once the optimized Jref is found, the same anti-reflection coating and back-reflector are applied to the structured solar cell device. In doing so, Jmax purely reflects the benefits of texturing the absorber material.
- Our literature analysis is applied to crystalline silicon (c-Si). Since the absorber material used for Jref also defines the values of JLL and JMB, the LTE remains applicable to any other technology or material. The LTE figure of merit is thus only limited by the availability of material parameters.
- Previous results indicate that the number of photogenerated electron-hole pairs is often found to be lower than the total measured absorption , which points to the presence of parasitic absorption. The short-circuit currents Jmax and Jref can therefore not be determined by the total simulated or measured absorption characteristic, but must refer to the sole active absorption, as already pointed out by several authors [12–14].
- To allow easier comparison of different approaches, we define the LTE for the total thickness of the absorber material. Other authors often refer to the effective thickness teff, which is a useful concept for comparing light trapping structures in terms of their material budget. However, the performance of a scatterer depends not only on the volume of the absorber material or on the geometry of the structures: the volume of material between the structures plays an equally important role as the scattering material itself. After all, scattering only happens at interfaces between materials of different refractive index, not in the homogenous material itself.
We applied the LTE defined in Eq. (5) to several c-Si solar cells found in the literature. While the majority of proposals are numerical simulations, some experimental results of hydrogenated microcrystalline silicon (µc-Si:H) devices are included in our study as well. However, since the material properties of µc-Si:H depend on both the growth conditions and the substrate layer, the LTEs of these structures are qualitatively assessed with the optical constants of c-Si. Figure 2 gives a review of the assessed structures, while the corresponding short-circuit currents are listed in Table 1, Table 2, and Table 3.
When the reference structure was not provided for the total thickness, we decided to use JMB as the reference in order to highlight light trapping performances.
We have introduced a new figure of merit to quantify the benefit of nanostructures for light trapping in solar cells, termed the light trapping efficiency (LTE). The LTE uses the short circuit current to assess the performance of a given light trapping design and compares it to an ideal Lambertian scatterer. The main difficulty in compiling a comprehensive overview of literature values is the large variety of parameters reported by authors, so we appeal to the community to become more consistent in how it reports efficiency enhancements and light trapping performance; we hope our work acts as an inspiration in this regard (Fig. 3). Similarly, we are not able to identify the “best” light trapping structure realised thus far, because it may be based on amorphous silicon or some other material without providing the relevant material properties. The fact that the performance of theoretical structures is significantly above that of experimental ones points to the need of theoretical studies to take “real” effects such as parasitic absorption better into account. In principle, the light trapping problem appears close to being solved, since we have identified a number of structures close to the ideal performance of LTE = 1. Naturally, demonstrating such high light trapping performance in real devices is another matter, which needs to be tackled next. Finally, the high performance of  and  highlights the need to achieve both high performance in light trapping and in antireflection, ideally together.
We like to add some specific comments to the quoted numbers and the selection of the papers.
Since the LTE is defined for the total thickness, we always recalculated JLL and JMB when authors referred to the effective thickness, e.g .
If the short-circuit currents were not quoted but the numerical absorption spectra provided , we used the free software tool Plot Digitizer from sourceforge.net in order to calculate the LTE of the structure. Since we always assumed an internal quantum efficiency equal to unity in numerical proposals, we excluded spectra where we suspected a parasitic influence. The criterion here was a high absorption near the wavelength bandgap of silicon: for a 300 µm thick layer, it takes a single-pass traversal to absorb 10% of the light, while in a 1 µm thick slab 20% would already be absorbed by a 99% reflective metal (assuming 2n2 “bounces” against the mirror).
Theoretical studies often use the optical constants provided by  or . We tried to take this into account for the LTE. However, if we found Jref too close to the current of a single-pass traversal, we recalculated the reference with the same optical constants, e.g .
The absorption enhancement of a thin-film with a Lambertian scattering layer was independently derived by  and . The same equations can be derived in a very intuitive way considering the attenuation of a propagating incoherent light ray in a lossy waveguide.
Our ansatz is schematically shown in Fig. 4, where we first distinguish between the top and bottom angle-averaged transmission coefficients T+ and T–. Since the Lambertian Scatterer is situated on the backside, the first traversal will be a single-pass transmission Tsp which is greater than T±. The effective back- and front reflectance, Rb and Rf, are then determined by the amount of light leaving the absorber into the adjacent layers.
The absorption can now be calculated via
When the Lambertian Scatterer is situated on top of the device, i.e. T → T–, and we assume that light is fully randomized, Tr = T+ ≈T–, we exactly find the previous derived equation from Brendel :
While M. Green  distinguishes between T+ and T–, we did not find any discrepancy in the outcoming result by using the assumption of T+ ≈T–. This also means that the absorption does not differ by much for dual or top Lambertian textures.
If the back-reflector is a perfect mirror, the front surface perfect anti-reflective and the incident medium air, we have Rb = 1 and Rf = 1 - 1/n2 leading to the absorption used in Eq. (1):
We wish to thank Daan Stellinga as well as Ken Xingze Wang of Stanford University and Olindo Isabella of Delft University for helpful discussions. This work was supported by the EU through Marie Curie Action FP7-PEOPLE-2010-ITN Project No. 264687 “PROPHET”.
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