# Evaluation of Voltage-Matched 2T Multi-Junction Modules Based on Monte Carlo Ray Tracing

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

_{1−x}Ga

_{x}Se

_{2}/Si (we will examine the case for x = 1, which is referred to as CGS in the rest of the work) and perovskite (PVK)/Si. In the next sections of the work, we will describe the optical simulation protocol and the assumptions about the electrical model that we considered to calculate the current–voltage (I-V) characteristics of each sub-cell and the multi-junction modules. Then, we will determine the optimal thicknesses of the semiconductors as well as of the transparent conductive oxide (TCO) layers for the case of the CdTe/Si module. Moreover, we will investigate the possibility of using modules with a flat Si surface to determine the contribution of Si surface texturization to the overall system efficiency. We will examine the dependence of the PVK/Si module efficiency on the bandgap energy of PVK. Lastly, we will determine the theoretical and practical PCE of the aforementioned modules in the optimized thickness configuration.

## 2. Optical Model

_{2}O

_{3}:Sn (ITO), we simplified the HJT structure by omitting the top and bottom thin hydrogenated amorphous Si layers. However, we considered the texturization of the Si cell, i.e., the presence of Si pyramids on the Si surface, since this has a major effect on light trapping. This point is described in more detail below.

- When the ray is traveling through an absorbing medium, an absorption probability $a\left(\lambda \right)=1-{e}^{-{\alpha}_{j}d}$ is calculated, with ${\alpha}_{j}$ being the absorption coefficient defined by Lambert–Beer’s Law and $d={l}_{j}/cos\theta $, where $\theta $ is the angle between $\overrightarrow{v}$ and the $\left(0,0,1\right)$ direction. A random number is generated to determine whether the ray is actually absorbed into the current layer or not. In the former case, a new iteration is initialized, whereas in the latter case, the position of the ray is updated and the iteration continues;
- When the ray reaches an interface, reflection and transmission probabilities are calculated from the Fresnel Equation [6] and a random number is generated to determine if the ray is reflected or transmitted. In the former case, the ray remains in the current layer and its direction is updated according to [17]; in the latter case, the layer is changed, and the ray direction is updated according to [17];
- When the ray approaches a pyramid, that is, when ${p}_{z}$ is in the range of the $z$ values of the points of the pyramid, the intersections between the $\overrightarrow{v}$ and the faces of the pyramid and of the unit cell are calculated, if present. The closest point $\overrightarrow{{p}^{\prime}}$ is selected, and, if the ray is traveling through an absorbing medium, a new absorption event is carried out as described above, with $d=\Vert \overrightarrow{p}-\overrightarrow{{p}^{\prime}}\Vert $. If the ray has not been absorbed and $\overrightarrow{{p}^{\prime}}$ lies on the pyramid, a new reflection/transmission event is carried out, whereas if $\overrightarrow{{p}^{\prime}}$ lies on the face of the unit cell, periodic boundary conditions are applied and the position is updated as $\overrightarrow{p}=\overrightarrow{{p}^{\prime}}+\widehat{u}\xb7s$, where $\widehat{u}$ is the unit vector perpendicular to the face pointing inwards and $s$ is the length of the edge of the base of the pyramid. The geometry of this step is summarized in Figure 2;
- If the ray travels from one surface of the Si layer to the other without being absorbed, its ${p}_{x}$ and ${p}_{y}$ will be randomized with respect to the pyramid of the new surface. This step reflects the fact that the position of pyramids obtained through chemical etching of Si is not correlated with the two surfaces of the layer;
- If the ray is in one of the two semi-infinite air layers and is traveling away from the system, it is considered either reflected (if $j=1$) or transmitted (if $j=11$), and a new iteration is initialized.

**Figure 2.**Interaction of the ray with a textured Si pyramid. Starting from the initial point P, the ray direction $\overrightarrow{v}$ intersects the pyramid at the points P′ and P″ as well as the boundary of the unit cell at P‴. The closest point to P is P′, which will be selected for the next step of the algorithm. $\overrightarrow{n}$ and $\overrightarrow{u}$ are the unit vectors perpendicular to the face of the pyramid and the unit cell, respectively.

_{1}(80 nm) and ITO

_{2}(100 nm), manufactured by Enel Green Power. Experimental data of reflectivity and transmissivity taken with a Bentham PVE300 for this structure were compared to the reflectivity and transmissivity spectra calculated through the present model. The results, reported in Figure 4, show that good agreement between the model and experimental data was found (Figure 4a), thus confirming the good accuracy of the proposed model, as well as the contribution of each layer in the sample system to the total absorbance (Figure 4b). More relevant deviations between model and data were found in the near-infrared region around 1000 nm, where an underestimate of absorbance was found. Such deviation is most likely due to relevant scattering effects occurring when the wavelength is close to the pyramid feature size, which cannot be accurately modeled using ray tracing, as shown in [23].

## 3. Electrical Model

_{Si}of 241.4 cm

^{2}in the Si sub-module, as this is a common case in commercial modules [29], and the number of cells in the top sub-module can then be obtained using

_{top}is equal to $72\xb7{S}_{Si}/{n}_{top}$. It is important to note that losses due to cell spacing in the module and the metallic grid shadowing are ignored.

_{Si}and S

_{top}are known, Equations (2) and (3) are used to calculate the I–V characteristics of a single cell; then, in each sub-module, the voltage in the I-V curve is multiplied by ${n}_{top}$ and ${n}_{Si}$, respectively, as the cells are connected in series with the assumptions that series resistance is negligible and that all cells are exactly equal in each sub-module. Finally, the I-V curves of the two sub-modules are summed in terms of current as these are connected in parallel, and the VM2T module’s efficiency in converting solar power to electrical power is then calculated as

## 4. Results and Discussion

#### 4.1. VM2T Module Optimization

_{1}, AZO

_{2}, ITO

_{1}and the interlayer of the VM2T module, we considered the case of the CdTe/Si module, since the refractive index of CdTe is close to the average of all the considered semiconductors for the top module, as can be seen from Figure 3. Rather than simulating all the possible combinations of the four parameters, we optimized one layer at a time, while keeping each optimal thickness fixed in the following steps (a similar approach was employed in [34,35]). Starting from the initial configuration ${l}_{AZ{O}_{2}}=100\mathrm{nm}$, ${l}_{IT{O}_{1}}=80\mathrm{nm}$, ${l}_{interlayer}=100\mathsf{\mu}\mathrm{m}$ and ${l}_{CdTe}=2\mathsf{\mu}\mathrm{m}$, we first varied the thicknesses of the TCO layers and the interlayer.

_{1}, AZO

_{2}, ITO

_{1}and interlayer thicknesses was the AZO

_{1}thickness.

_{1}, AZO

_{2}, ITO

_{1}and interlayer thicknesses optimized as shown in Figure 5 and Figure 6, it was possible to focus on the optimization of the TCS thickness. This variable was allowed to span from 0.1 µm to 4 µm, and the results are reported in Figure 7, showing the VM2T system efficiency under STC as a function of TCS thickness. From the inspection of Figure 7, it is evident that the PCE increased with the TCS thickness, though the PCE increased with respect to a decrease in thickness greater than 1 µm. Since a minority of carrier diffusion lengths in thin-film PV technologies are generally of the order 1 of μm or below [37,38], and given the change in slope at 1 µm in Figure 7, we assumed 1 µm to be the maximum realistically possible TCS thickness, as the increase in absorbance in thicker films would be balanced out by the increased recombination effects.

#### 4.2. VM2T PVK/Si Module Analysis

_{1}, AZO

_{2}, ITO

_{1}and the interlayer reported in the previous section, we varied the PVK bandgap energy from 1.6 eV to 2 eV (with a fixed thickness of 2 µm) for the cases of the textured Si module and the flat Si module. Figure 9 reports the results of these calculations. In Figure 9a, it can be observed that in both cases, the optimal bandgap energy was 1.85 eV, which is quite different from the value commonly reported in the literature of 1.7 eV [41]. However, this value is usually obtained through ab initio calculations, where the absorbances of the materials are assumed to be one, neglecting optical effects altogether. On the contrary, the simulations showed that an accurate evaluation of the performance of a photovoltaic system must include an optical model that considers the refractive indices of all materials involved. Moreover, it has to be noted that the difference in PCE between the modules with textured and flat Si surfaces increased with the PVK bandgap energy, in accordance with what was stated in the previous section; in fact, it can be seen that PVK’s contribution to the overall PCE decreased with an increasing bandgap, whereas the contribution of Si (and of its texture) increased.

## 5. State-of-the-Art Module Efficiencies

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Layers of the VM2T multi-junction module. The top sub-module is flat, whereas the Si and ITO layers of the bottom sub-module are textured.

**Figure 4.**Comparison between experimental and simulated absorbances, reflectances and transmittances of an ITO/c-Si/ITO cell (

**a**). Stacked area chart of the simulated absorbances of each layer in the test sample (

**b**).

**Figure 5.**Optimization of the TCO layers and interlayer in the CdTe/Si module. The figure reports the SQ PCEs obtained by varying the thickness of the first AZO layer (

**a**), the second AZO layer (

**b**), the first ITO layer (

**c**) and the interlayer (

**d**).

**Figure 6.**Optimization of the TCO layers and interlayer in the CdTe/Si module with flat ITO and Si top surfaces. The figure reports the SQ PCEs obtained by varying the thickness of the first AZO layer (

**a**), the second AZO layer (

**b**), the first ITO layer (

**c**) and the interlayer (

**d**).

**Figure 7.**SQ PCEs of the GaAs/Si, CdTe/Si and CGS/Si VM2T modules with textured Si sub-module (

**a**) and flat Si sub-module (

**b**).

**Figure 8.**Real (

**a**) and imaginary (

**b**) parts of the refractive indices of PVK for bandgaps ranging from 1.6 eV to 2 eV obtained from Kramers–Kronig relations. Absorption coefficients (

**c**) of PVK for the same bandgaps [7]. Values relative to the literature material are included for comparison. The legend applies to all sub-figures.

**Figure 9.**PCE as a function of PVK bandgap at the SQ limit for the VM2T PVK/textured Si and the VM2T PVK/flat Si module at a fixed PVK thickness of 2 µm (

**a**). SQ PCE of the PVK/textured Si module and the PVK/flat Si module and each sub-module for different thicknesses of the 1.85 eV bandgap PVK (

**b**). TCO layers and interlayer thicknesses were fixed to the values obtained from the optimization processes for multi-junction modules with textured and flat Si surfaces in Section 4.1.

**Table 1.**Bandgap energies, open-circuit voltages and technology development coefficients (TDCs) of the semiconductors considered in this work. For CGS and PVK, the bandgap of the record cells reported in the literature are also listed. It must be noted that for these materials, TDC calculations were based on the bandgap and voltages of the literature material, while the coefficient was assumed to be the same for the bandgap considered in this work.

Semiconductor | Bandgap_{this work} (eV) | Bandgap_{literature} (eV) | V_{OC,SA} (V) | V_{OC,SQ} (V) | TDC |
---|---|---|---|---|---|

Si | 1.12 | - | 0.738 | 0.860 | 0.858 |

GaAs | 1.42 | - | 1.1272 | 1.144 | 0.985 |

CdTe | 1.5 | - | 0.875 | 1.215 | 0.720 |

CuIn_{1-x}Ga_{x}Se_{2} | 1.7 (x = 1) | 1.08 | 0.734 | 0.822 | 0.893 |

PVK | 1.85 | 1.67 | 1.213 | 1.374 | 0.883 |

**Table 2.**Comparison between number of TCS cells and PCEs obtained in the SQ limit and for state-of-the-art cells for the VM2T modules with textured Si. PCEs of each TCS single-junction record cell as reported in [43] are included.

Module | ${\mathit{n}}_{\mathit{T}\mathit{C}\mathit{S},\mathit{S}\mathit{Q}}$ | ${\mathit{n}}_{\mathit{T}\mathit{C}\mathit{S},\mathit{S}\mathit{A}}$ | PCE_{TCS,SA} (%) | PCE_{VM2T,SQ} (%) | PCE_{VM2T,SA} (%) |
---|---|---|---|---|---|

GaAs/Si | 50 | 44 | 29.1 | 30.51 | 29.45 |

CdTe/Si | 48 | 57 | 21.0 | 34.53 | 25.24 |

CGS/Si | 42 | 40 | 23.35 | 36.07 | 31.74 |

PVK/Si | 38 | 37 | 23.7 | 36.77 | 34.16 |

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**MDPI and ACS Style**

Corso, R.; Leonardi, M.; Milazzo, R.G.; Scuto, A.; Privitera, S.M.S.; Foti, M.; Gerardi, C.; Lombardo, S.A.
Evaluation of Voltage-Matched 2T Multi-Junction Modules Based on Monte Carlo Ray Tracing. *Energies* **2023**, *16*, 4292.
https://doi.org/10.3390/en16114292

**AMA Style**

Corso R, Leonardi M, Milazzo RG, Scuto A, Privitera SMS, Foti M, Gerardi C, Lombardo SA.
Evaluation of Voltage-Matched 2T Multi-Junction Modules Based on Monte Carlo Ray Tracing. *Energies*. 2023; 16(11):4292.
https://doi.org/10.3390/en16114292

**Chicago/Turabian Style**

Corso, Roberto, Marco Leonardi, Rachela G. Milazzo, Andrea Scuto, Stefania M. S. Privitera, Marina Foti, Cosimo Gerardi, and Salvatore A. Lombardo.
2023. "Evaluation of Voltage-Matched 2T Multi-Junction Modules Based on Monte Carlo Ray Tracing" *Energies* 16, no. 11: 4292.
https://doi.org/10.3390/en16114292