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Abstract
Silicon-based tandem solar cells and modules are complex systems that require optical modeling for the optimization towards highest efficiencies. The fact that such devices typically incorporate surface structures of different optical regimes poses high requirements to the involved simulation tools. The OPTOS formalism is ideally suited to deal with such complexity. Within this work OPTOS is extended in order to calculate the layer resolved absorptance in silicon-based tandem solar cells and module stacks. After describing the relevant mathematical details, a good agreement between OPTOS absorptance simulation results and EQE measurements of the current 33.3% record efficiency III-V on silicon two-terminal tandem solar cell is found. Furthermore, a detailed loss analysis is performed for an exemplary perovskite silicon solar cell with and without module encapsulation. The comparison reveals a lower photocurrent density for the module stack due to increased reflectance and absorption in the EVA.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Silicon-based tandem solar cells can overcome the efficiency limit of single junction solar cells. This is especially relevant as pure silicon solar cells are approaching their efficiency limit and further improvement is difficult to realize experimentally. An outstanding example of such a tandem solar cell is the two-terminal record device by Cariou et al. which consists of two III-V solar cells on top of a silicon bottom cell reaching an efficiency of 33.3% [1]. Perovskite silicon tandem solar cells have not reached such high efficiencies yet. However, record values have risen quickly up to 23.6% [2] over the last years.
Compared to the silicon single junction case, tandem solar cells consist of additional layers and materials. Optical simulation can be a valuable technique to support the design of tandem devices and optimize their performance. Due to the potentially complex layer configuration suitable simulation methods include as many as possible of the following capabilities:
- i. Coherent as well as incoherent treatment of planar films or sheets.
- ii. The consideration of structures with dimensions that are large compared to the wavelength and allow an approximate treatment with geometrical optics.
- iii. The consideration of structures with dimensions that are in the range of the wavelength or below and require a wave optical treatment.
- iv. The incorporation of encapsulation effects.
- v. An efficient variation of the angle of incidence.
- vi. The calculation of the layer resolved absorptance.
Most optical simulation methods for silicon solar cells are based on raytracing in combination with a thin film solver such as the transfer matrix formalism (TMM) and thereby comply with aspects i and ii [3–5]. Widely used is the tool Sunsolve by PV-lighthouse which has an easy to use web interface and also fulfills aspects iv, v and vi [6]. However, all pure raytracing tools cannot easily include small scale structures such as diffractive gratings (aspect iii). This can be extremely important as the III-V on silicon record shows. One of the key ingredients to reach the efficiency of 33.3% is a diffractive grating at the rear of the silicon cell, which enhances the light trapping properties of the device.
The two matrix-based formalisms, OPTOS by Fraunhofer ISE [7,8] and Genpro by Delft University [9] can combine interfaces with large and interfaces with small scale structures since each interface is modelled independently with a suitable approach (e.g. RCWA for diffractive gratings, and ray tracing for pyramidal textures) as first step. The calculated redistribution properties of each interface are stored as matrices and used to calculate the absorptance, reflectance or transmittance of the investigated solar cell in a second step. Parameter variations of the interface structures can be time consuming, but once the light redistribution matrices are calculated, variations of the silicon absorber thickness or the angle of incidence can be calculated within seconds or minutes. With OPTOS also encapsulation effects (aspect iv) can be considered and results for a yield analysis for different locations including many different angles of incidence (aspect v) can be calculated [10,11].
This work covers an extension of the OPTOS formalism to calculate the layer resolved absorptance of the full device (aspect vi). This is essential for tandem solar cells in order to determine the absorptance and hence the current generation in all subcells and to perform a detailed optical loss analysis. After presenting the mathematical formulation of this extension, OPTOS simulation results are compared to measurements of the 33.3% III-V on silicon tandem solar cell. Further capabilities of the method are demonstrated at an encapsulated perovskite silicon solar cell including a planar module front side, random pyramids at the silicon front side and a diffractive grating with metal reflector at the rear.
2. Layer resolved absorptance
The calculation of the layer resolved absorptance using the OPTOS formalism can be discussed without loss of generality at the exemplary system depicted in Fig. 1(b). This two-terminal tandem solar cell incorporates a perovskite top cell consisting of a thin film stack on a silicon bottom cell with 250 µm bulk thickness. The silicon front side features a random pyramid texture and the rear side a diffractive grating. The light redistribution properties for reflection and transmission are stored for a predefined set of angle channels (θi, φj) in precalculated redistribution matrices (a detailed description of angle channels and matrices for different interface structures can be found in [7,8,10]). The angular light distributions after every interaction are vectors (vi) which can be calculated solely based on redistribution matrices and the incident light distribution vector v0 [7].
Fig. 1 Sketches of an exemplary perovskite silicon tandem solar cell as investigated in this work. a) Thin film stack of the perovskite top solar cell. b) Complete perovskite silicon solar cell with random pyramid texture at the front and diffractive grating at the rear side. The arrows indicate the light propagation between the interfaces. The angular light redistribution is stored in the vectors vi. c) Complete simulated module stack including ethyl-vinyl-acetate (EVA) as encapsulation material and a planar front anti-reflective coating.
It is also possible to sum certain light distribution vectors of the different passes vi in order to gain the total light distribution incident on a specific interface. For example, one can sum all vi incident to the front side from within the silicon to obtain the total angular light distribution incident onto that interface:
This representation simplifies the following discussion.Calculating the redistribution matrices for the perovskite top cell means to determine reflection and transmission of the layer stack for all incident angle channels. Using thin film solvers such as the scattering matrix formalism [12,13] or the transfer matrix formalism [14], also the absorptance (Absj) of every layer j can be calculated for every incident angle. This information is stored in an absorptance matrix, which has the form described in Eq. (2) for light travelling upwards to the front interface of the exemplary system of Fig. 1(b). The rows correspond to the angle channels, the columns to the different layers of the stack.
The absorptance matrix for downwards travelling light has the same format. Note that Eq. (2) is a simplified formula showing only the polar angle dependence. For all simulations within this work both the polar as well as the azimuth angle are used in order define the angle channels.The total absorptance in a specific layer j consists of the absorptance of the light incident from the top side (air) and light incident from the bottom (silicon) which can be calculated as follows:
This extension can be easily integrated into the existing formulation of the OPTOS formalism and the additional simulation time is negligible. It can also be included when calculating redistribution matrices of effective interfaces, as described in [10]. Thereby, also encapsulated tandem solar cells can be modeled and optical loss analyses for each single layer can be performed.3. Optical analysis of experimental III-V on silicon tandem record cell
For a comparison between simulation and measurement results, the III-V on silicon two-terminal record solar cell with 33.3% efficiency is investigated. It consists of a GaInP top cell, a GaAs middle cell and a 280 µm thick silicon bottom cell, an anti-reflective coating, a window layer and several tunnel junctions. The world record device has a planar front side and a diffractive grating integrated at the rear side. In this investigation also a reference device with planar front and rear side is included. More details about the layer thicknesses, grating fabrication and characterization can be found in [1].
OPTOS absorptance simulation results and EQE measurements of the record as well as the reference solar cell are compared in Fig. 2.
Fig. 2 Left: Measured external quantum efficiency (EQE) and simulated absorptance results of the III-V on silicon tandem solar cell with the current two-terminal record efficiency of 33.3% [1]. In the simulation a dielectric grating with additional 25% parasitic absorption was used as simplified representation of the structured metal grating in the experimental device. Right: Sketch of the modeled solar cell.
The OPTOS simulation of the reference system with planar front and rear (black - dotted) fits well to the measured EQE data (black – solid). This demonstrates that the general approach and implementation discussed in section 2 can describe experimental results with high accuracy. Modeling the diffractive grating of the record device, which incorporates a structured metal grating as reflector, is computationally very demanding. As approximation, redistribution matrices of a silicon grating with surrounding air and a planar metal rear reflector were used instead (see right side of Fig. 2). 25% parasitic absorption was added in the simulation at each interaction of the light with this interface. This accounts for parasitic absorption effects at the structured metal reflector in the experimental device. For more details of this simplified simulation approach and its application to silicon single junction solar cells see [15]. It leads also for the tandem device discussed here to a good agreement of simulated absorption and measured EQE results.
4. Optical analysis of a perovskite silicon tandem solar cell and module stack
4.1 Material choice and structure definition
A perovskite silicon tandem solar cell is used in this work as exemplary system to demonstrate further capabilities of the extended OPTOS formalism. Due to the diffractive rear side grating the system cannot be modeled with geometrical optics. Modeling the front side pyramids with wave optical methods would require immense computational resources due to the large structure dimensions. However, for the use in OPTOS, redistribution matrices of all interfaces can be calculated with raytracing and a thin film solver for the pyramid structure and perovskite cell. Rigorous coupled wave analysis (RCWA) is used for the diffractive grating and again a thin film solver for the planar module front side. The random pyramids were modeled as described in [8,16,17]. The parameters of the diffractive rear side grating, e.g. a period of 1 µm, were chosen based on experimentally realized samples described in [18]. The grating described there led to an EQE enhancement for silicon single junction cells with planar front side, which was in good agreement with simulation results. The refractive index data of the perovskite cell thin film stack, depicted in Fig. 1(a), is taken from literature: magnesium fluoride (MgF2) [19] as anti-reflective coating, indium tin oxide (ITO) [20] as charge transport material at top and bottom, Spiro-OMeTAD [21] as hole contact material, MaPbI3 [22] as perovskite absorber and C60 [23] as electron contact layer. Data for evaporated titanium oxide (TiO2) as electron contact layer was determined via spectral ellipsometry [24]. The data for the encapsulation made of EVA was taken from [25] and for the silicon bulk from [26].
4.2 Optimization of the perovskite layer stack
Before determining the redistribution matrices for their use in OPTOS, the layer thicknesses of the textured perovskite cell were optimized using a simplified system. A planar perovskite cell with half infinite air or EVA at the top and half infinite silicon at the bottom was modeled and absorptance as well as transmittance were calculated via thin film solvers. In order to estimate reflectance and transmittance of the pyramidal front side, the two most relevant incident angles, 54.7° for the first interaction and 15.8° for the second one, were used as incidence angles onto the planar stack. The results were combined to an approximated photocurrent density according to Eq. (4) and (5):
with incident spectrum I as well as an assumed absorption of transmitted photons (AoTP) of the silicon cell. The AoTP was determined via OPTOS simulations of a similar layer stack with the same materials but estimated layer thicknesses. Subsequently, the minimum of these estimated photocurrent densities was maximized using an evolutionary algorithm [27], since the lower current limits the device in a two-terminal tandem configuration.The results of this layer thickness optimization are summarized in Table 1. For all contact and charge transport layers, namely ITO, Spiro-OMeTAD, C60 and TiO2, the optimal thicknesses are equal to the minimum thicknesses, which were set in the evolutionary algorithm.
Table 1. Layer thicknesses of the perovskite stack in solar cell and module stack configuration
4.3 Loss analysis at cell level
As next step, redistribution matrices were calculated for all required interfaces based on the optimized layer thicknesses. The calculation time is very short for planar interfaces but can be time consuming for random pyramids since many angles of incidence have to be considered for the different wavelengths of interest. The subsequent OPTOS simulations led to an optical loss analysis including all layers of the device, as depicted in Fig. 3 for the tandem solar cell described above without encapsulation. The direct reflection is very small due to the pyramidal front side which leads to multiple light interactions with the surface. In the low wavelength range parasitic absorption in the Spiro-OMeTAD is a relevant effect. The ITO layers absorb strongly in the long wavelength range, which might look surprising at first. However, front side pyramids and diffractive rear side grating lead to light paths within the silicon bulk with many bounces between front and rear side. Therefore, the light interacts several times with the front side and parasitic absorption in the respective layers increases.
Fig. 3 Loss analysis of a perovskite silicon tandem solar cell with a matched photocurrent density of 19.6 mA/cm2. The layer resolved analysis reveals that Spiro-OMeTAD and the two ITO layers show strong parasitic absorption.
4.4 Loss analysis at module stack level
The same analysis for the encapsulated system is depicted in Fig. 4. The EVA encapsulation generally leads to a higher parasitic absorption in the low wavelength range. This results in a slightly smaller absorptance of the Spiro-OMeTAD layer. For the system investigated here, an increase of the photocurrent density could be reached by choosing a transparent conductive oxide (TCO) with less parasitic absorption. However, investigating the influence of different TCO material parameters to the photocurrent density of the perovskite and silicon subcells is beyond the scope of this work. Furthermore, in an experimental realization the TCO has to guarantee a sufficient level of conductivity in order to prohibit resistive losses.
Fig. 4 Loss analysis of an encapsulated perovskite silicon tandem solar cell. Absorption in the EVA encapsulation layer reduces the photocurrent density to 19.0 mA/cm2 in the perovskite top cell and 18.9 mA/cm2 in the silicon bottom cell.
The module stack configuration shows reduced photocurrent densities of 19.0 mA/cm2 in the perovskite top cell and 18.9 mA/cm2 in the silicon bottom cell compared to the unencapsulated case. This is due to absorption in the 500 µm thick EVA layer as well as a slight reflection increase which is caused by the additional interface at the module front. As one can see, the thickness of the perovskite layer has to be reduced in the module case on order to achieve current matching. Hence it is important to investigate optimal layer thicknesses for the full module structure and not for the tandem cell alone, in order to obtain the best performance in the module case.
5. Conclusion
Within this work, the OPTOS formalism was extended to calculate the layer resolved absorptance. It was shown to be capable of modeling silicon-based tandem solar cells and module with a complex combination of surface structures. The comparison of OPTOS absorptance simulations and EQE measurements of the record III-V on silicon two-terminal tandem solar cell shows a good agreement. The layer resolved absorptance results for an exemplary perovskite silicon tandem solar cell with large scale random pyramids at the silicon front and a small scale diffractive grating at the rear reveals strong parasitic absorption in the ITO layers of the perovskite top cell. When comparing cell and module stack results, the absorption in the Spiro-OMeTAD was slightly reduced in the encapsulated case due to absorption in the EVA. For the exemplary system investigated here, most effort has to be put into optimizing the ITO material properties in order to reduce parasitic absorption while keeping a sufficient level of conductivity. These insights demonstrate the capabilities of the OPTOS formalism for the modeling of silicon-based tandem solar cells.
Funding
German Federal Ministry of Economic Affairs and Energy (0324037A “Persist”, 0324247 “Potasi”); German Research Foundation (“Oposit”).
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