# A Unified Description of the Electrical Properties with Complex Dynamical Patterns in Metal Halide Perovskite Photovoltaics

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Device Modeling

#### 2.1. Advanced Surface Polarization Model

#### 2.2. On the Impedance Spectroscopy Dynamics

## 3. Results and Discussion

_{2}/m-TiO

_{2}/perovskite/spiro-OMeTAD/Au was selected as representative for a perovskite solar cell with a nominal composition of Cs

_{0.05}MA

_{0.15}FA

_{0.80}Pb

_{1.05}(Br

_{0.15}I

_{0.85})

_{3}for the active layer. All fabrication procedures and electrical characterization are described in the Supplementary Material.

^{2}(i.e., 1 Sun). In Figure 3a, it is clearly observed that there are now two loops in the current–voltage curve, where the current in forward and backward makes a crossing pathway. This effect indicates a transition from regular to inverted hysteresis [19,31,32] as the voltage increases, caused by the appearance of the chemical inductor in the impedance spectra. Next, we analyze the transformation region around this change in behavior. Nyquist plots at relatively low voltages manifest as two neatly distinguished capacitive semicircles (Figure 3b). Nevertheless, at a voltage domain, a transformation of or change in the equivalent circuit occurs in which the low-frequency positive arc vanishes and gives rise to an inductive loop in the positive complex half-plane, as is shown the Figure 3c. Note that the term “transformation” refers to the simplified version of the equivalent circuit of Figure 1b since the electrical components of the model are not present in the entire voltage range. The values calculated from the impedance data corresponding to Figure 3b,c, using the equivalent circuit of Figure 1b, are detailed in Table 1. As in other studies [29,30], the value of the geometrical capacitance, constant and independent of the voltage, is on the order of hundreds of nanofarads (for high frequencies).

## 4. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Novel dynamic electrical model for perovskite-based solar cells proposed in this work, where a fractional-order capacitor is introduced by a polarization mechanism. (

**b**) Equivalent circuit representation, valid for impedance interpretation, obtained by applying a small AC modification of a DC steady state.

**Figure 2.**(

**a**) Dark voltage–current curves of perovskite solar cells under study here performed at different scan rates from 5 to 1000 mV/s. (

**b**) Complex impedance plot measured under dark conditions and at 1 V. The inset of (

**b**) shows the nonideal inductive loop in detail.

**Figure 3.**(

**a**) Cyclic voltammetry of Cs

_{0.05}MA

_{0.15}FA

_{0.80}Pb

_{1.05}(Br

_{0.15}I

_{0.85})

_{3}perovskite solar cells measured at 25 mV/s under illumination of 1 Sun. (

**b**,

**c**) Representative impedance evolution from 1 to 1.3 V, concerning the transition from capacitive to inductive effects.

**Figure 4.**Evolution of time constants and constant-phase element exponents obtained from the impedance parameters throughout the voltage range.

Bias Voltage (V) | ${\mathit{C}}_{\mathbf{g}}$ $\left(\text{\xb5}\mathbf{F}\text{\xb7}{\mathbf{cm}}^{-2}\right)$ | ${\mathit{R}}_{\mathbf{rec}}$ $\left(\mathsf{\Omega}\text{\xb7}{\mathbf{cm}}^{2}\right)$ | ${\mathit{R}}_{\mathbf{C}}$ $\left(\mathsf{\Omega}\text{\xb7}{\mathbf{cm}}^{2}\right)$ | ${\mathit{C}}_{\mathbf{\alpha}}$ $\left(\mathbf{mF}\text{\xb7}{\mathbf{cm}}^{-2}\text{\xb7}{\mathit{s}}^{\mathit{\alpha}-1}\right)$ | $\mathbf{\alpha}$ | ${\mathit{R}}_{\mathbf{L}}$ $\left(\mathsf{\Omega}\text{\xb7}{\mathbf{cm}}^{2}\right)$ | ${\mathit{L}}_{\mathit{\varphi}}$ $(\mathbf{H}\text{\xb7}{\mathbf{cm}}^{2}\xb7{\mathit{s}}^{\mathit{\varphi}-1})$ | $\mathit{\varphi}$ |
---|---|---|---|---|---|---|---|---|

1 | 0.28 | 33.63 | 23.53 | 4.68 | 0.62 | – | – | – |

1.1 | 0.27 | 8.32 | 142.71 | 0.23 | 0.68 | – | – | – |

1.2 | 0.32 | 3.94 | – | – | – | 9.01 | 0.73 | 0.71 |

1.3 | 0.51 | 1.74 | – | – | – | 2.24 | 0.35 | 0.77 |

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**MDPI and ACS Style**

Hernández-Balaguera, E.; Martin-Martin, D.
A Unified Description of the Electrical Properties with Complex Dynamical Patterns in Metal Halide Perovskite Photovoltaics. *Fractal Fract.* **2023**, *7*, 516.
https://doi.org/10.3390/fractalfract7070516

**AMA Style**

Hernández-Balaguera E, Martin-Martin D.
A Unified Description of the Electrical Properties with Complex Dynamical Patterns in Metal Halide Perovskite Photovoltaics. *Fractal and Fractional*. 2023; 7(7):516.
https://doi.org/10.3390/fractalfract7070516

**Chicago/Turabian Style**

Hernández-Balaguera, Enrique, and Diego Martin-Martin.
2023. "A Unified Description of the Electrical Properties with Complex Dynamical Patterns in Metal Halide Perovskite Photovoltaics" *Fractal and Fractional* 7, no. 7: 516.
https://doi.org/10.3390/fractalfract7070516