Estimating the performance of solar cells with luminescent down-shifting layers

Technological developments for improving the performance of conventional solar cells have become a topic of great interest in recent years. For instance, solar concentrators, new anti-reflective coatings, and Luminescent Down-Shifting Layers (LDS), among different techniques have been used in the past. The latter is an attractive option because an LDS layer has the property of increasing the photon flux density in the appropriate wavelength range on top of a cell device with the possibility of increasing the photo-current density. Then, in this work we focus on the development of a theoretical model to determine the cell´s illumination current density, considering the modified solar spectrum, and taking in account the modified spectral reflectance and transmittance at the upper layers when an optimized LDS layer is inserted on a solar cell. The correct selection of such a layer for a specific solar cell would increase its performance due to the enhanced photon density in the absorption region for which the solar cell has the highest quantum efficiency. As an example, it is shown that a Lu3Al5O12:Ce3+ layer on top of a CdTe solar cell might cause an efficiency increase of around 21%.


Introduction
Research on new materials for their use in the optoelectronics industry has made it possible to design devices that achieve high performance at low production costs. Specifically, in the photovoltaic field there are at least three (so called) cell generations under development, so that the photovoltaic energy has become competitive with other sources of electricity in the market [1]. However, it is necessary to continue the solar cell progress for sharing an even a bigger portion of the electrical energy sector [2,3].
Solar cells should absorb most of the solar spectrum for converting it into electrical power. For example, tandem solar cells include absorber layers with different bandgaps for absorbing different regions of the spectrum [4], optimizing the conversion efficiency in each region. Similarly, the use of Luminescent Down-Shifting (LDS) layers allows the modification of the photon flux density to be absorbed by the device, so that the modified spectrum is converted into electrical power with an improved efficiency [5][6][7]. This option has been one of the less explored, although experimental and theoretical reports have shown that it may increase efficiency significantly [8].
Theoretical studies for solar cells that include an LDS layer calculate the external quantum efficiency, the photocurrent, and the cell efficiency [9]. For example, in references [10,11] Song, Zhang and Zhu have proposed a model for solar cells with an LDS layer. Unfortunately, their model is not correct, even dimensionally, and some of the parameters involved in the model has not been explained appropriately, nor how these parameters are determined, so that their results cannot be reproduced by other researchers. Additionally, their model did not take in account the effect of the optical transmittance due to the LDS layer itself and it considered only average values for the reflectance because they did not do calculations for the cell´s spectral reflectance (and transmittance). Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. Therefore, in this work, a new theoretical model is shown for calculating the effect due to the modified AM1.5 solar spectrum by an LDS layer, including also the modified spectral transmittance and reflectance by the presence of the LDS layer itself on top of the solar cell. In this model, the optical properties of the different materials that are part of the cell, including the LDS layer, are considered to achieve more complete and exact calculations.
The developed model can be applied to different kind of solar cells with appropriate LDS layers. For example, Cadmium Telluride (CdTe) cells have achieved record efficiencies above 22% [12], but a larger efficiency increase has become technologically challenging. For this reason, the model described in this work is used to calculate the effect of an LDS (Lu 3 Al 5 O 12 :Ce 3+ ) layer on CdTe solar cells. Two configurations, LDS/glass/SnO 2 /CdS/CdTe/Ag and glass/LDS/SnO 2 /CdS/CdTe/Ag are compared, and it is shown that in both cases the CdTe cell can improve its efficiency around 21% when using Lu 3 Al 5 O 12 :Ce 3+ as an LDS layer.

Model
The inclusion of an LDS layer on a solar cell should increase the photon flux density in a wavelength range that can be absorbed by the cell with a high conversion efficiency. The LDS layer will absorb high energy photons coming from the solar spectrum, where the cell typically has a low quantum efficiency, and transform them into lower energy photons which will be absorbed and converted into electricity with a higher efficiency by the solar cell. Then, the available power for photovoltaic conversion in a solar cell can be expressed as the sum of two terms, one due to the direct solar spectrum on the cell (reduced by the reflectance and transmittance losses on the upper layers) plus the additional photon density due to the power shifting by the LDS layer. In other words, the available photon flux l P , ( ) can be written as follows: where f l s ( ) corresponds to the AM1.5 solar spectrum photon flux density. The reflectance R(λ) and transmittance T(λ) due to the upper layers are determined through an optical matrix method reported previously by our group [13,14]. For these calculations it is essential to have measured or reported data in the literature on the extinction coefficient (k) and refractive index (n) for each of the layers that constitute the device.
The additional photon density f l M ( ) due to the LDS layer is: where q is the magnitude of the electron´s charge and h l int ( ) is the cell´s internal quantum efficiency. The limits are the minimum wavelength (λ min ) for photons in the respective solar spectrum and the maximum wavelength (λ g ) to be absorbed by the absorber material in the device.

Case study for CdTe solar cells
A practical application of the model described in the previous section is using a (Lu 3 Al 5 O 12 :Ce 3+ ) layer on a CdS/CdTe solar cell with a conventional structure (Glass/SnO 2 /CdS/CdTe/contact). We use two configurations for including the LDS layer. The first one corresponds to the LDS layer between the glass and SnO 2 layer and the second is for the LDS layer on top of the front glass, as depicted in figure 1(a) and (b), respectively.
It is worth noticing that in the first case, a non-conventional fabrication process would be needed (deposition of the LDS layer before the SnO 2 layer in the fabrication of the solar cell) while in the second case, the LDS layer could be deposited on finished modules, but then these layers would be exposed to the environmental conditions. Hence, both configurations would require technological challenges to be solved before the LDS inclusion in this kind of solar cells. Despite this fact to be solved yet, in this work we focus our attention on the possible efficiency improvement due to the presence of an LDS layer in these devices.
Using values of n and k as a function of wavelength for each of the materials [15-17] that constitute the CdTe device (see figure 2), the R(λ) and T(λ) calculation were made using an optical matrix method [13,14], as mentioned above. For the Lu 3 Al 5 O 12 :Ce 3+ LDS layer, the n(λ) and k(λ) spectra, as reported by other authors, were used [18,19]. Figure 3 shows R(λ) and T(λ) and the results obtained, including the Lu 3 Al 5 O 12 :Ce 3+ layer for a conventional CdTe cell. In all cases, the thicknesses of the glass/SnO 2 /CdS/CdTe/Ag layers had constant values. The LDS layer thickness was varied between 100 nm and 1000 nm. Figure 3 shows the spectra only for the optimum 200 nm LDS layer on top of the glass and the optimum 300 nm LDS layer behind the glass, respectively, as will be explained below.
To determine the Lu 3 Al 5 O 12 :Ce 3+ optimum thickness, the reflectance R̅ and transmittance T̅ averages weighed on the solar spectrum were calculated, so that the product -* R T 1 ( ̅ ) ̅ becomes maximum, for the respective configuration (see figure 4).
After determining the optimum LDS layer thickness for each configuration, equation (4) was used to obtain the photon flux density when the LDS layer is included, for each configuration. For this purpose, b l E ( ) due to the Lu 3 Al 5 O 12 :Ce 3+ layer was obtained from reference [18] and it is shown as a dotted line in figure 5. Finally, the average LDS layer optical efficiency h á ñ DS was assumed to be 0.5, considering that about half of the photons are emitted towards and half outwards the cell´s surface, respectively.
The modified photon flux density spectra shown in figure 5, for the two configurations, show a significant increase in the region between 450 nm and 600 nm. In this region, the cell´s external quantum efficiency will become higher than 100%. Therefore, a better solar cell performance can be expected. The internal quantum efficiency h l int ( ) for a typical (CdTe) solar cell was already calculated by our group and reported in a previous work [20]. All the required parameters such as carrier mobilities, lifetimes, absorber thickness, interface recombination velocities, etc for the internal quantum efficiency calculation can be consulted in that reference [20].
The photocurrent density was determined using expression (5) with λ min = 280 nm and λ g = 815 nm for CdTe. A significant change in the photocurrent density is obtained for the cells with the LDS layers, in both configurations, as shown in table 1. Assuming that there is no significant modification for the Voc and FF without any LDS layer, a conversion efficiency increase, of the order of 21%, can be expected when the LDS layers are included in the cell structure.    . Calculated reflectance and transmittance spectra for the different CdTe solar cell structures, with optimum Lu 3 Al 5 O 12 :Ce 3+ layer thickness, using an optical matrix model [13,14]. associated to the presence of the LDS layer itself. The model was applied to calculate the modified photocurrent density of CdTe solar cells with a Lu 3 Al 5 O 12 :Ce 3+ LDS layer. It was shown that a photocurrent density and efficiency increase, around 21%, can be expected, so that it could be an alternative approach for improving this kind of solar cells in the future.

Data availability statement
All data that support the findings of this study are included within the article (and any supplementary files).