# Optimization of N-PERT Solar Cell under Atacama Desert Solar Spectrum

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Model Validation at STC

_{λ}) of the metallized solar cell, shown in Figure 2, was obtained using a Perkin Elmer 950 Spectrophotometer [3]. See the Appendix A for the full parameter list.

- (1)
- The model assumes flat surfaces; thus, a correction factor must be introduced to match the short current density. However, the increase in surface area is accompanied by an increment in the saturation current density ${J}_{0}$ due to recombination. Fell et al. [1] considered both issues and defined correction factors (${f}_{corr}$) for different solar cells. In this work, the calculated short current density under STC conditions, ${J}_{sc,cal}$, was multiplied by ${f}_{corr}=1.17$ for comparison with the measured short current density, ${J}_{sc,meas}$.
- (2)
- The model does not consider metal induced recombination. Therefore, the open circuit voltage ${V}_{oc}$ may be higher than the experimental result. These differences were quantified and fully described by Edler et al. [29]. They quantified the reduction in the open circuit voltage $\Delta {V}_{oc}$ of n-PERT solar cells by varying the metal fraction ${f}_{met}$ and quantifying the dark saturation current density at the metal/semiconductor interfaces. For the metal fraction of n-PERT solar cells in this work (0.099 and 0.052 for the front and rear side, respectively), the estimation indicates that the ${V}_{oc}$ can decrease 45 mV due to the front side metallization and 35 mV due to the rear side metallization. It is pointed out that the effect of series and shunt resistances, ${R}_{ser}$ and ${R}_{shunt}$, is observable in the shape of the IV curve, and thus, on the $FF$ and ${P}_{mpp}$ [30]. Based on the referenced experimental work, metal induced recombination can be applied after the model is solved, by subtracting $\Delta {V}_{oc}$ for the corresponding metal fraction. In addition, ohmic losses will affect ${P}_{mpp}$ and $FF$. For power, ${P}_{mpp}={P}_{mpp,0}-{R}_{ser}{I}_{mpp}{}^{2}$, where ${P}_{mpp,0}$ is the power without resistance effects. For the fill factor, $FF=F{F}_{0}\left(1-{r}_{ser}\right)$, where $F{F}_{0}$ is the fill factor without resistive effects, ${r}_{ser}$ is the normalized series resistance to the characteristic resistance (${r}_{ser}={R}_{ser}/{R}_{CH}$ and ${R}_{CH}={V}_{oc}/{I}_{sc}$) [31].

#### 2.2. Determination of the Optimal Solar Cell Parameters

- Selection: We calculated the value of the cost function $g$ of all points ${x}_{i}^{n}$, and it was denoted as ${g}_{i}^{n}=g\left({x}_{i}^{n}\right)$, with $i=1,\dots {N}_{p}$. For each ${x}_{i}^{n}$, a probability ${p}_{i}^{n}$ to be selected, was assigned. The probability can be written in terms of the cost function, ${p}_{i}^{n}=p\left({g}_{k}^{n}\right)$. Once the probability was computed, $2{N}_{p}$ elements called parents were randomly chosen, and they were denoted as ${y}_{i}^{n}$, with $i=1,\dots 2{N}_{p}$. This procedure assumed that the region of the points with the lowest value of $g$ is explored with a larger frequency.
- Crossover: We created ${N}_{p}$ elements called children and denoted as ${e}_{i}^{n}$, with $i=1,\dots {N}_{p}$, from the values of the parents ${y}_{i}^{n}$ by considering a random point included in the segment defined by two parents. This step was intended to explore a zone included between two parents’ points and determine if there was a better element.
- Mutation: We modified randomly some components of the ${e}_{i}^{n}$ elements. The goal was to explore some areas of the search space randomly. In addition, this step allowed for escape from possible local minima, which may attract too many elements of the population.
- Elitism: We aimed to ensure that the convergence of the GA was always decreasing, that is, the value of $g$ of the best element from each generation was decreasing from one generation to another. Thus, we directly copied the best element from the previous generation ${X}^{n}$ and it was denoted as ${\overline{x}}^{n}$.

## 3. Theory

- i.
- Net charge density relation

- ii.
- Transport equations

- iii.
- Continuity equations

^{−1}), and $z$ the spatial coordinate. Assuming that each photon leads to an electron–hole pair, dismissing reflection, the generation rate ${G}_{0}$ (in cm

^{−3}s

^{−1}) is computed as the derivative of $\varphi \left(\lambda ,z\right)$ with respect to $z$ and integrating over $\lambda $ (Equation (7)). A similar process was used by Chowdhury et al. [36].

## 4. Results

#### 4.1. Measurements and Simulation of the Solar Cell at Standard Conditions

^{2}. Using the value of the series resistance ${R}_{ser}=0.45\mathsf{\Omega}{\mathrm{cm}}^{2}$ from the experimental measurements, the following results were obtained.

#### 4.2. Optimal Solar Cell Parameters

_{E}= 650 nm or d

_{E}= 200 nm), a shorter time is required [38]. In the case of the thickness of the BSF, the same reasoning as above follows. Regarding the thickness of the silicon wafer (cell thickness), optimal values are smaller than the initial one. The use of thinner wafers implies a more efficient use of material (cost reduction) and the possibility of better heat management under illumination [39]. Additionally, the voltage benefits from a lower thickness [40].

^{+}) in this case [42] (where the “+” denotes higher doping concentration). This result for the BSF is inverse to that at the emitter because photons are mostly absorbed in the emitter and base. Thus, limiting doping concentration at the rear side is not necessarily due to a lower excess of minority carrier density compared to the front side, allowing a thinner and heavier doped BSF. Studies regarding the optimal thickness for a phosphorus doped layer, such as the BSF of the n-PERT solar cell, indicate that for low doping concentration, the thickness is higher compared to that at higher doping concentration, where thickness needs to be limited to avoid recombination [43]. Stem and Sid [43] found that the optimal thickness for the P-doped layer is below 1 µm for doping concentrations from 5 × 10

^{19}cm

^{−3}to 1 × 10

^{20}cm

^{−3}, reaching highest efficiency in the range of 0.2 to 0.4 µm.

## 5. Discussion

#### 5.1. Solar Spectrum

#### 5.2. Metallization

_{3}+ HF + HNO

_{3}treatment to remove the fingers’ glass and silver and allow the etched SiNx area to be visible. It was found that the etched area can vary between 51% and 67% depending on paste composition. The more aggressive paste led to poorer solar cell performance, expressed in the efficiency. In both cases, this etching process meant that the passivation underneath the metal contact was removed, leading to an enhanced recombination at the metal–Si interface [29]. Penetration of Ag crystallites is the main contributor to the current transport between Ag and c-Si [46]. Based on 60 atomic force microscopy (AFM in Figure 7c) measurements at different locations, it turned out that Ag crystallites reached a penetration depth up to 40 ± 5 nm. As shown in Table 4, the emitter thickness of around 200 nm is large enough to avoid substantial recombination due to the metallization.

## 6. Conclusions

^{+}nn

^{+}structure and, thus, a family of cases, where p

^{+}refers to the emitter, n is the base, and n

^{+}is the back surface field (BSF). Additionally, the model can work for the case of monofacial and bifacial solar cells under any illumination, which means, only front, only rear, or simultaneous front and rear side illumination.

_{sc}, +5.7% for the power P

_{mpp}, and +5.4% for the efficiency η, with a loss in the open circuit voltage V

_{oc}of 1%.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

**Table A1.**Summary of input parameters. Note that the shown ${d}_{E},\text{}{d}_{cell},\text{}{d}_{BSF},\text{}{N}_{E},\text{}{N}_{B},\text{}\mathrm{and}\text{}{N}_{BSF}$ values were used for the mesh and validation study. However, these parameters were control variables in the optimization step.

Input Parameter | Value |
---|---|

Emitter thickness, ${d}_{E}$ (nm) | 650 |

Wafer thickness, ${d}_{cell}$ (µm) | 180 × 10^{3} |

BSF thickness, ${d}_{BSF}$ (nm) | 450 |

Simulated area, $A$ (cm^{2}) | 1 |

Relative dielectric constant, ${\epsilon}_{r}$ | 11.7 |

Electron affinity, $\chi $ (eV) | 4.05 |

Bandgap, ${E}_{g}$ (eV) | 1.12 |

Effective conduction band density, ${N}_{c}$ (cm^{−3}) | (T/300[K]) ^{3/2} × 1.04 × 10^{19} [cm^{−3}] |

Effective valence band density, ${N}_{v}$ (cm^{−3}) | (T/300[K])^{3/2} × 2.8 × 10^{19} [cm^{−3}] |

Electron mobility, ${\mu}_{n}$ (cm^{2}V^{−1}s^{−1}) | Arora model |

Hole mobility, ${\mu}_{p}$ (cm^{2}V^{−1}s^{−1}) | Arora model |

Front-side surface doping, ${N}_{E}$ (cm^{−3}) | 2.44 × 10^{19} |

Base doping concentration, ${N}_{B}$ (cm^{−3}) | 8.44 × 10^{14} |

Rear-side surface doping, ${N}_{BSF}$ (cm^{−3}) | 6.17 × 10^{19} |

c-Si density, $\rho $ (kg/m^{3}) | 2329 |

Auger Recombination coefficient for electrons, ${C}_{n}$ (cm^{6}s^{−1}) | 2.80 × 10^{−31} |

Auger Recombination coefficient for holes, ${C}_{p}$ (cm^{6}s^{−1}) | 9.90 × 10^{−32} |

Direct band-to-band recombination coefficient, $C$ (cm^{3}s^{−1}) | 4.73 × 10^{−15} |

Tau trap for electrons and holes, ${\mu}_{n},{\mu}_{p}$ (ms) | 1 |

**Table A2.**Summary of input parameters for the Arora model for the mobility of electrons ${\mu}_{n}$ and holes ${\mu}_{p}$, taken from [34]. The Arora model equations are shown below the table.

Input Parameter | Value |
---|---|

Electron mobility reference, ${\mu}_{n,0}$ (cm^{2}/(Vs) | 1252 |

Hole mobility reference, ${\mu}_{p,0}$ (cm^{2}/(Vs) | 407 |

Electron mobility reference minimum, ${\mu}_{n,min}^{ref}$ (cm^{2}/(Vs) | 88 |

Hole mobility reference minimum, ${\mu}_{p,min}^{ref}$ (cm^{2}/(Vs) | 54.3 |

Electron reference impurity concentration, ${N}_{n,0}^{ref}$ (1/cm^{3}]) | 1.26 × 10^{17} |

Hole reference impurity concentration, ${N}_{p,0}^{ref}$ (1/cm^{3}) | 2.35 × 10^{17} |

Alpha coefficient, ${\alpha}_{0}$ | 0.88 |

Mobility reference minimum exponent, ${\beta}_{1}$ | −0.57 |

Mobility reference exponent, ${\beta}_{2}$ | −2.33 |

Impurity concentration reference exponent, ${\beta}_{3}$ | −2.33 |

Alpha coefficient exponent, ${\beta}_{4}$ | −0.146 |

Reference temperature, ${T}_{ref}$ (K) | 300 |

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**Figure 2.**Spectral reflectance of the n-PERT solar cell [3].

**Figure 3.**Mesh independence study of the main output JV parameters as a function of the mesh elements for a 1D model.

**Figure 5.**Current–voltage curve of the optimized and non-optimized n-PERT solar cell at the AM1.5G and AM1.08 spectra.

**Figure 6.**Spectral response of an n-PERT solar cell, measured for the front and rear sides under 1 sun.

**Figure 7.**(

**a**) SEM image of a silver finger of the n-PERT solar cell with a width around 50 µm and height of 10 µm after a single print. This structure was removed for the interface analysis. (

**b**) Image of the SiNx etched area due to metallization; the light gray areas correspond to the etched areas (for further details, see [27]). (

**c**) AFM image and depth profile of Ag imprints left behind after the chemical treatment.

Name | Value | Description |
---|---|---|

${d}_{cell}$ | 180 µm | Solar cell thickness |

${d}_{E}$ | 0.65 µm | Emitter depth |

${d}_{BSF}$ | 0.45 µm | Thickness of BSF |

${N}_{E}$ | 2.44 × 10^{19} cm^{−3} | Emitter surface conc. |

${N}_{B}$ | 8.436 × 10^{14} cm^{−3} | Base doping |

${N}_{BSF}$ | 6.17 × 10^{19} cm^{−3} | BSF surface conc. |

${\tau}_{n}$ | 1.5 ms | SRH Carrier lifetime |

${\tau}_{p}$ | 1.5 ms | SRH Carrier lifetime |

${f}_{met}$ | 0.052 | Metal fraction front side |

JV Measurement | JV Simulation | |||
---|---|---|---|---|

Parameter | Front | Rear | Front | Rear |

${J}_{sc}\left(\mathrm{mA}/{\mathrm{cm}}^{2}\right)$ | 39.2 ± 0.03 | 34.6 ± 0.03 | 39.2 | 34.2 |

${V}_{oc}\left(\mathrm{mV}\right)$ | 653.1 ± 2 | 649.7 ± 2 | 646.4 | 654.6 |

${P}_{mpp}\left(\mathrm{W}\right)$ | 4.9 ± 0.02 | 4.3 ± 0.09 | 5.3 | 4.6 |

$FF(\%)$ | 78.3 ± 0.2 | 78.2 ± 0.16 | 78.7 | 78.7 |

$\eta (\%)$ | 20 ± 0.08 | 17.6 ± 0.1 | 20.0 | 18.0 |

Solar Cell Parameters | Initial Values | Range |
---|---|---|

${d}_{E}$ (nm) | 650 | 50–750 |

${d}_{cell}$ (nm) | 180 × 103 | 150 × 103–200 × 103 |

${d}_{BSF}$ (nm) | 450 | 50–750 |

${N}_{E}$ (cm^{−3}) | 2.44 × 10^{19} | 1 × 10^{19}–1 × 10^{20} |

${N}_{B}$ (cm^{−3}) | 8.44 × 10^{14} | 1 × 10^{14}–5 × 10^{15} |

${N}_{BSF}$ (cm^{−3}) | 6.16 × 10^{19} | 1 × 10^{19}–5 × 10^{20} |

GA Parameters | Value |
---|---|

Population size | 70 |

Generations | 110 |

Stopping criterium | 10 |

Mutation probability | 10% |

Non-Optimized | Optimized | Increment | ||||
---|---|---|---|---|---|---|

Parameter | STC AM1.5G | ATA AM1.08 | STC AM1.5G | ATA AM1.08 | STC AM1.5g | ATA AM1.08 |

${J}_{sc}\left(\mathrm{mA}/{\mathrm{cm}}^{2}\right)$ | 39.2 | 42.2 | 40.7 | 44.3 | 3.7% | 4.9% |

${V}_{oc}\left(\mathrm{mV}\right)$ | 646.4 | 647.0 | 641.7 | 640.1 | −0.7% | −1.1% |

${P}_{mpp}\left(\mathrm{W}\right)$ | 5.3 | 5.7 | 5.5 | 6.1 | 4.3% | 5.7% |

$FF(\%)$ | 78.7 | 78.9 | 79.1 | 80.1 | 0.5% | 1.5% |

$\eta (\%)$ | 20.0 | 21.6 | 20.8 | 22.7 | 4.3% | 5.4% |

Description | Parameter | Exp. Values | AM1.5G | AM1.08 |
---|---|---|---|---|

Emitter thickness | d_{E} (nm) | 650 | 200.2 | 201 |

Cell thickness | d_{cell} (nm) | 180 × 10^{3} | 154.2 × 10^{3} | 165.1 × 10^{3} |

BSF thickness | d_{BSF} (nm) | 450 | 330 | 250 |

Emitter doping | N_{E} (cm^{−3}) | 2.44 × 10^{19} | 9.89 × 10^{19} | 9.36 × 10^{19} |

Base doping | N_{B} (cm^{−3}) | 8.44 × 10^{14} | 9.83 × 10^{14} | 9.81 × 10^{14} |

BSF doping | N_{BSF} (cm^{−3}) | 6.16 × 10^{19} | 3.87 × 10^{20} | 4.92 × 10^{20} |

n-PERT Front (AM 1.5G) | n-PERT Rear (AM 1.5G) | n-PERT Front (AM 1.08) | n-PERT Rear (AM 1.08) | |
---|---|---|---|---|

Measured J_{sc} (mA/cm^{2}) | 39.2 | 34.6 | X | X |

Calculated J_{sc} via COMSOL (mA/cm^{2}) | 39.2 | 34.2 | 44.3 | X |

Calculated J_{ph} (mA/cm^{2}) | 38.9 | 34.3 | 42.2 | 37.1 |

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**MDPI and ACS Style**

Ferrada, P.; Marzo, A.; Ferrández, M.R.; Reina, E.R.; Ivorra, B.; Correa-Puerta, J.; Campo, V.d.
Optimization of N-PERT Solar Cell under Atacama Desert Solar Spectrum. *Nanomaterials* **2022**, *12*, 3554.
https://doi.org/10.3390/nano12203554

**AMA Style**

Ferrada P, Marzo A, Ferrández MR, Reina ER, Ivorra B, Correa-Puerta J, Campo Vd.
Optimization of N-PERT Solar Cell under Atacama Desert Solar Spectrum. *Nanomaterials*. 2022; 12(20):3554.
https://doi.org/10.3390/nano12203554

**Chicago/Turabian Style**

Ferrada, Pablo, Aitor Marzo, Miriam Ruiz Ferrández, Emilio Ruiz Reina, Benjamin Ivorra, Jonathan Correa-Puerta, and Valeria del Campo.
2022. "Optimization of N-PERT Solar Cell under Atacama Desert Solar Spectrum" *Nanomaterials* 12, no. 20: 3554.
https://doi.org/10.3390/nano12203554