# Analysis of Losses Associated with Series Resistance (Rs) in Simple-Structured c-Si Solar Cells

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## Abstract

**:**

## 1. Introduction

^{2}[7]. This analysis ranges from calculating the series resistance based on semiconductor physics parameters to determining Rs by applying the straight-line slope technique to the I-V curve of each manufactured device.

## 2. Series Resistance (Rs) Components and c-Si Structure

#### 2.1. Influence on the Efficiency of Ohmic Resistances

#### 2.2. Components of the Series Resistance (Rs)

R1 | Back metal contact |

R2 | Metal-semiconductor contact across the entire back surface |

R3 | Semiconductor material (base) |

R4 | Resistance of the emitter between two grid fingers |

R5 | The metal-semiconductor contact of the grid finger |

R6 | Grid finger resistance |

R7 | Busbar resistance |

$q{\phi}_{Bn},\text{}q{\phi}_{Bp}$ | Height of the metal-semiconductor barrier |

$q{\phi}_{m}$ | Work function of the metal |

$q{\chi}_{S}$ | Work function of the semiconductor |

${E}_{g}$ | Bandgap of the semiconductor |

^{6}A/m

^{2}K

^{2}), and m* = the mass of the electron.

#### 2.3. Single-Junction c-Si Solar Cells

_{2}, and the upper and lower metal (Al) contacts [7].

#### Influence of Sheet Resistance on the Efficiency of the Solar Cell

_{SH}, indirectly affects the efficiency of the solar device, depending on the magnitude it can reach and whether it is a uniformly doped substrate or a non-uniformly doped one [11].

_{SH}can be determined according to:

_{SH}can be calculated as:

_{n}= electron mobility, and C(x) = the concentration profile.

_{SH}of the substrate based on the junction depth between the materials. This implies that R

_{S}will also undergo changes according to these parameters and, in turn, will modify the final cell efficiency as shown in Table 3.

#### 2.4. Top Contact Grid

_{S}) is the nature of the metallic contacts, as they are responsible for facilitating the flow of electrons from the cell to the load [12]. Figure 5 illustrates the configuration of the front grid, which commonly consists of a main bus and transverse collector bars, known as “fingers” [13].

#### 2.5. Configuration of Resistances for the Calculation of Rs

_{Eq}) is shown in the new resistive network of Figure 8. Through this updated arrangement and by the Equation (9) is applied to simplify the circuit, the value of the series resistance (Rs) of the cell is determined.

## 3. Results

_{SHSUB}), and using the four-point technique in the case of the emitter (R

_{SHE}).

_{SHE}and R

_{SHSUB}, depending on the junction depth of each manufacturing process, I–IV, are presented in Table 5.

_{SHE}and R

_{SHSUB}vary when adjusting xj. By applying Equation (6) with a resistivity of 5 Ω-cm for the substrate, the thickness of t

_{sub}will be modified based on the junction depth and, consequently, R

_{SHSUB}will change.

_{SHSUB}using the four-point technique allows us to determine the resistivity of the substrate (p-type), even without forming the p-n junction. Table 6 shows the comparison between experimentally obtained R

_{SHSUB}and theoretically calculated R

_{SHSUB}by applying Equation (6). For these calculations, a value of ρ

_{SUB}of 8.5 Ω-cm was used, generating a discrepancy in R

_{SHSUB}of less than 3%. Nevertheless, it is evident that the application of Equation (6) is effective for calculating R

_{SH}in situations where the substrate is uniformly doped.

_{SH}for each side of the cell and R

_{SHSUB}and R

_{SHE}for the substrate and emitter, respectively. These calculations take into account that these components are affected by the junction depth in each process. Table 7 presents the results obtained for each of these components.

_{SHSUB}and R

_{SHE}vary with the junction depth, directly impacting those components dependent on these parameters.

_{SHSUB}but depends on the thickness of the p-region (t

_{PR}), the value of which is modified according to the depth of the emitter, ranging between 0.87 µm and 0.60 µm.

_{SH}. Their resistance stays constant and is only presented in the first section of Table 7 for a depth of 0.87 µm.

_{Eq}is obtained, it is possible to apply Equation (9) to obtain the series resistance of the solar cell.

_{SH}on total Rs.

_{OC}region, where series resistance exerts a notable influence. Furthermore, Table 8 provides a comprehensive account of the percentage variance between the aforementioned calculations and those derived using the parameter extraction technique.

_{SC}). This phenomenon can be attributed to the decrease in the distance that charge carriers (electrons and holes) generated by sunlight must travel before being captured by the electrical contacts of the cell. As a result, recombination losses are reduced, and the current generated by the cell increases [7,15].

## 4. Discussion

_{SH}, which is also part of the final Rs. Additionally, a circuit is proposed that models the components of Rs, considering it as series resistance without assuming that all its components are in series as well.

## 5. Conclusions

_{SH}) are an integral part of the manufacturing process. The consideration of R

_{SH}is critical, as elevated values of this component lead to increased Rs values. In [9], there is a proposal for the use of substrates with a thickness of 100 μm, as such substrates achieve heightened efficiencies.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Losses in solar cells [2].

**Figure 2.**Influence of ohmic resistances on the I-V curve of solar devices: (

**a**) series resistance; (

**b**) parallel resistance.

**Figure 3.**Components of series resistance (Rs) [2].

**Figure 9.**Obtaining Rs by the parameter extraction method described in [14]. (

**a**) xj = 0.87 μm; (

**b**) xj = 0.75 μm; (

**c**) xj = 0.70 μm; (

**d**) xj = 0.60 μm.

**Table 1.**Definition of equations to calculate each component of Rs [2].

Rs Component | Equation |
---|---|

R1 | ${R}_{1}=\frac{1}{6}{\rho}_{met}\frac{{L}_{BC}}{{{t}_{met}W}_{BC}}$ |

R2 | ${R}_{2}=\frac{\sqrt{{R}_{\mathrm{S}\mathrm{H}\mathrm{S}\mathrm{U}\mathrm{B}}{\rho}_{CSUB}}}{{L}_{BC}}coth\left({W}_{BC}\sqrt{\frac{{R}_{\mathrm{S}\mathrm{H}\mathrm{S}\mathrm{U}\mathrm{B}}}{{\rho}_{CSUB}}}\right)$ |

R3 | ${R}_{3}={\rho}_{SUB}\ast {t}_{PR}\ast {A}_{CELL}$ |

R4 | ${R}_{4}=\frac{{R}_{SHE}}{{L}_{f}}\frac{S}{6}$ |

R5 | ${R}_{5}=\frac{\sqrt{{R}_{\mathrm{S}\mathrm{H}\mathrm{E}}{\rho}_{CE}}}{{L}_{f}}coth\left({W}_{f}\sqrt{\frac{{R}_{\mathrm{S}\mathrm{H}\mathrm{E}}}{{\rho}_{CE}}}\right)$ |

R6 | ${R}_{6}=\frac{1}{3}{\rho}_{met}\frac{{L}_{B}}{{t}_{m}{W}_{B}}$ |

R7 | ${R}_{7}=\frac{1}{6}{\rho}_{met}\frac{{W}_{f}}{{t}_{m}{W}_{f}}$ |

**Table 2.**Description of the parameters used for the calculation of Rs components [2].

Parameter | Description |
---|---|

${\rho}_{met}$ | Resistivity of the metal |

${\rho}_{SUB}$ | Resistivity of the substrate |

${\rho}_{CE}$ | Specific contact resistivity at the emitter |

${\rho}_{CSUB}$ | Specific contact resistivity of the substrate |

${R}_{SHSUB}$ | Substrate sheet resistance |

${R}_{SHE}$ | Emitter sheet resistance |

${W}_{f}$ | Finger width |

${L}_{f}$ | Finger length |

${W}_{BC}$ | Back contact width |

${L}_{BC}$ | Back contact length |

${W}_{B}$ | Busbar width |

${L}_{B}$ | Busbar length |

${t}_{met}$ | Metal thickness |

${t}_{SUB}$ | Substrate thickness |

${t}_{PR}$ | Thickness of the p region |

${A}_{CELL}$ | Solar cell area |

**Table 3.**Comparison between junction depth and solar cell efficiency [7].

Process | xj (μm) | η (%) |
---|---|---|

I | 0.87 | 5.8 |

II | 0.75 | 6.5 |

III | 0.70 | 7.1 |

IV | 0.60 | 8.2 |

Parameter | Value | Units |
---|---|---|

${\mathsf{\rho}}_{\mathrm{m}\mathrm{e}\mathrm{t}}$ | 2.82 × 10^{−8} | Ω-m |

${\mathsf{\rho}}_{\mathrm{S}\mathrm{U}\mathrm{B}}$ | 5–15 | Ω-cm |

${\mathrm{W}}_{\mathrm{f}}$ | 100 | μm |

${\mathrm{L}}_{\mathrm{f}}$ | 1.9 | μm |

${\mathrm{W}}_{\mathrm{B}\mathrm{C}}$ | 1 | cm |

${\mathrm{L}}_{\mathrm{B}\mathrm{C}}$ | 1 | cm |

${\mathrm{W}}_{\mathrm{B}}$ | 300 | μm |

${\mathrm{L}}_{\mathrm{B}}$ | 1 | cm |

${\mathrm{t}}_{\mathrm{m}\mathrm{e}\mathrm{t}}$ | 5 | μm |

${\mathrm{t}}_{\mathrm{S}\mathrm{U}\mathrm{B}}$ | 300 | μm |

${\mathrm{A}}_{\mathrm{C}\mathrm{E}\mathrm{L}\mathrm{L}}$ | 1 | cm^{2} |

Manufacturing Process | xj (μm) | R_{SHE} (Ω/sq) | R_{SHSUB} (Ω/sq) |
---|---|---|---|

I | 0.87 | 16.43 | 167.12 |

II | 0.75 | 18.08 | 167.36 |

III | 0.70 | 24.71 | 167.08 |

IV | 0.60 | 29.71 | 167.11 |

Process | Experimental | Theoretical |
---|---|---|

t_{sub} | 300 μm | 300 μm |

ρ_{SUB} | 8.7376 Ω cm | 8.5 Ω cm |

R_{SHSUB} | 291.25 Ω/sq | 283.33 Ω/sq |

xj = 0.87 μm | |

Rs Component | Value (Ω) |

R1 | 4.8 × 10^{−3} |

R2 | 6.8854 |

R3 | 1.9443 × 10^{−9} |

R4 | 1.0727 |

R5 | 0.4250 |

R6 | 0.4656 |

R7 | 144 × 10^{−6} |

xj = 0.75 μm | |

Rs Component | Value (Ω) |

R2 | 6.004 |

R3 | 1.9451 × 10^{−9} |

R4 | 1.1804 |

R5 | 0.6091 |

xj = 0.70 μm | |

Rs Component | Value (Ω) |

R2 | 5.9140 |

R3 | 1.9454 × 10^{−9} |

R4 | 1.6133 |

R5 | 0.7931 |

xj = 0.60 μm | |

Rs Component | Value (Ω) |

R2 | 5.4271 |

R3 | 1.9461 × 10^{−9} |

R4 | 1.94 |

R5 | 0.9771 |

η (%) | Rs Theoretical (Ω) | Rs Extraction Model (Ω) | Diff (%) |
---|---|---|---|

5.8 | 8.261 | 8 | 3.14 |

6.5 | 7.803 | 6 | 23.10 |

7.1 | 7.380 | 6 | 18.69 |

8.2 | 6.944 | 5 | 27.99 |

xj (μm) | I_{SC} (A) | V_{OC} (V) | P_{MAX} (W) |
---|---|---|---|

0.87 | 21.85 × 10^{−3} | 0.50 | 5.79 × 10^{−3} |

0.75 | 21.92 × 10^{−3} | 0.50 | 6.75 × 10^{−3} |

0.70 | 24.71 × 10^{−3} | 0.50 | 7.09 × 10^{−3} |

0.60 | 27.54 × 10^{−3} | 0.52 | 8.24 × 10^{−3} |

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**MDPI and ACS Style**

Heredia-Rios, M.J.; Hernandez-Martinez, L.; Linares-Aranda, M.; Moreno-Moreno, M.; Méndez, J.F.
Analysis of Losses Associated with Series Resistance (Rs) in Simple-Structured c-Si Solar Cells. *Energies* **2024**, *17*, 1520.
https://doi.org/10.3390/en17071520

**AMA Style**

Heredia-Rios MJ, Hernandez-Martinez L, Linares-Aranda M, Moreno-Moreno M, Méndez JF.
Analysis of Losses Associated with Series Resistance (Rs) in Simple-Structured c-Si Solar Cells. *Energies*. 2024; 17(7):1520.
https://doi.org/10.3390/en17071520

**Chicago/Turabian Style**

Heredia-Rios, Manuel J., Luis Hernandez-Martinez, Monico Linares-Aranda, Mario Moreno-Moreno, and Javier Flores Méndez.
2024. "Analysis of Losses Associated with Series Resistance (Rs) in Simple-Structured c-Si Solar Cells" *Energies* 17, no. 7: 1520.
https://doi.org/10.3390/en17071520