Semiconductors have become essential in photovoltaic (PV) technologies due to their ability to directly convert sunlight into electrical energy. Among various PV technologies, perovskite solar cells (PSCs) have drawn significant attention for their high power conversion efficiency, low manufacturing cost, and compatibility with low-temperature, solution-based fabrication methods [1] [2] . Unlike conventional silicon-based solar cells, PSCs offer easier processing and the potential for higher efficiencies, with reports of over 25% efficiency in recent years [1] [3] . However, key challenges such as environmental instability, degradation under moisture and heat, and current-voltage hysteresis still hinder large-scale commercialization [4] [5] .

The performance and stability of PSCs are strongly influenced by charge transport dynamics within their layered structure, typically consisting of a Hole Transport Layer (HTL), a perovskite absorber, and an Electron Transport Layer (ETL). Charge transport is governed by drift due to internal electric fields and diffusion driven by carrier concentration gradients [6] . Moreover, interfacial effects, including ion migration and charge accumulation, contribute to hysteresis and performance losses [7] [8] . To gain insight into these mechanisms, one-dimensional drift-diffusion models are widely used, incorporating effects such as light-induced carrier generation and potential barriers at interfaces [9] - [11] . This study investigates the steady-state behaviour of a three-layer PSC by numerically solving the coupled drift-diffusion equations, aiming to better understand charge carrier dynamics and their impact on overall device performance.

Modelling solar cell operation involves solving a set of coupled nonlinear partial differential equations that describe key physical mechanisms such as photogeneration, charge carrier transport via drift and diffusion, recombination, and interfacial effects across multiple material layers. These models incorporate numerous parameters; including carrier mobilities, lifetimes, dielectric constants, and built-in potentials which contribute to their mathematical complexity and sensitivity to boundary and initial conditions [12] [13] . Despite this complexity, many existing studies simplify the modelling by assuming steady-state conditions and neglecting temporal effects, thus overlooking critical phenomena such as hysteresis, ion migration, and transient responses under dynamic illumination or biasing conditions [5] [14] . This research was explicitly solving time dependence of the charge transport equations, enabling a more realistic analysis of dynamic processes in perovskite solar cells. Although the results presented are steady state, the time-dependent drift-diffusion equations were solved to capture the cell transient behaviour before the cell reached steady state. This study contributes toward SDG 7: Affordable and Clean Energy by providing insights that support the development of high-efficiency, stable perovskite solar cells through improved numerical modelling and device optimization.

Modelling the perovskites solar required the understanding of the working principle of the solar cell. The basic mechanisms of the solar cell consist of different types of processes, that are the generation process, transportation process, and collection process. The operation of perovskite solar cell is shown in Figure 1 .

Figure 1 shows the mechanism of a perovskite solar cell. In the perovskite solar cell, the structure of the perovskite layer acts as a blend phase layer ( $-\frac{L}{2}<x<\frac{L}{2}$) is sandwiched between the p-type donor layer ( $-l_1-\frac{L}{2}<x<-\frac{L}{2}$), and the n-type acceptor layer ( $\frac{L}{2}<x<l_2+\frac{L}{2}$). The sunlight is absorbed in the perovskite layer that has a highly ordered crystalline structure comprised of distinct conduction band edges, ${\hat{\varphi }}_n$, and valence band edges, ${\hat{\varphi }}_p$ that are divided by a band gap. Furthermore, the organics materials of the donor phase (p-type) and acceptor phase (n-type) have no distinct band structure and are amorphous. LUMO portrays the conduction band, and the energy of the acceptor is denoted as ${\varphi }_n$ while HOMO portrays the valence band, and the energy of the donor is denoted as ${\varphi }_p$. The electrons are excited into the LUMO, leaving a hole in the HUMO, resulting in the conduction process due to the excited electrons and hole movement between the HUMO and LUMO respectively on the adjacent molecules. This process is commonly known as hopping between shallows and traps for highly localized energy wells.

This study employs numerical techniques to solve the drift-diffusion equations governing charge transport in perovskite solar cells, with a focus on the Method of Lines (MOL). These equations, which include both partial and ordinary differential equations (PDEs and ODEs), need to be solved over time to demonstrate how the system changes [10] [15] . While commercial solvers such as SCAPS-1D, COMSOL, and Ion Monger are commonly used, they are limited by assumptions and predefined settings [16] [17] . This study focuses specifically on modeling charge transport within the perovskite absorber layer. However, the numerical framework developed here is general and can be extended to include adjacent layers such as the electron transport layer (ETL) and hole transport layer (HTL), allowing for full device simulations in future work.

Method of Lines (MOL) is a semi-discretization method in which spatial derivatives in Equations (1) - (3) are discretized (assume that the diffusion coefficients of the holes and electrons are constants) to convert the governing PDEs into a system of ordinary differential equations (ODEs) that can be solved using established time integration methods [18] . Discretize Equations (1) - (3) to obtain

$\begin{array}{c}\frac{\partial p_i}{\partial t}=\frac{D_p}{h^2}\left[p_{i+1}-2p_i+p_{i-1}+\frac{q}{kT}\frac{\left(p_i+p_{i+1}\right)\left({\phi }_{i+1}-{\phi }_i\right)-\left(p_i+p_{i-1}\right)\left({\phi }_i-{\phi }_{i-1}\right)}{2}\right]\\ +b\left(n_i{p}_i-N_i^2\right)\end{array}$ (6)

$\begin{array}{c}\frac{\partial n_i}{\partial t}=\frac{D_n}{h^2}\left[n_{i+1}-2n_i+n_{i-1}-\frac{q}{kT}\frac{\left(n_i+n_{i+1}\right)\left({\phi }_{i+1}-{\phi }_i\right)-\left(n_i+n_{i-1}\right)\left({\phi }_i-{\phi }_{i-1}\right)}{2}\right]\\ +b\left(n_i{p}_i-N_i^2\right)\end{array}$ (7)

$0=\frac{1}{h^2}\left[{\phi }_{i+1}-2{\phi }_i+{\phi }_{i-1}\right]-\frac{q}{ϵ}\left(n_i-p_i\right)$ (8)

This results in a system of coupled ODEs, where each equation (Equations (6) - (8)) corresponds to a discretized spatial point. These equations describe the time evolution of variables such as electron and hole densities and the electrostatic potential across the device layers;

$M\frac{\partial u_i}{\partial t}=Au+f\left(x,t\right)$(9)

The $u_{N\times 1}$ is the solution vector, $M_{N\times N}$ represents the time-dependent mass matrix, $A_{N\times N}$ denotes the differentiation matrix, and the $f\left(x,t\right)$ is the generation and recombination of charge density of holes and electrons, while for the Poisson equation, it is the difference between electrons and holes.

$M={\left[\begin{array}{ccc}{M}_p& 0& 0\\ 0& M_n& 0\\ 0& 0& 0\end{array}\right]}_{3N\times 3N},A={\left[\begin{array}{ccc}{A}_p& 0& 0\\ 0& A_n& 0\\ 0& 0& A_{\varphi }\end{array}\right]}_{3N\times 3N},f\left(x,t\right)={\left[\begin{array}{c}{f}_p\\ f_n\\ f_{\varphi }\end{array}\right]}_{3N\times 1}$ (10)

where p is the system for holes, n is the system for electrons, and $\varphi$ is the system for potential difference.

One major advantage of the MOL is its flexibility and compatibility with various stiff and non-stiff ODE solvers. In this study, the resulting ODE system (Equations (9) - (10)) is solved using MATLAB’s ODE15s solver, which is well-suited for stiff systems and those involving singular mass matrices. The MOL method also allows for straightforward incorporation of boundary and initial conditions, and it scales efficiently with increasing spatial resolution, making it an ideal choice for simulating the dynamic behavior of complex, multilayer solar cell structures. The set of parameters for the perovskite solar cell used in this model is tabulated in Table 1 .

The simulation result for the unsteady state condition at the blend phase layer (perovskite layer) is shown in dimensional form. The comparison of the simulation result obtained and the experimental simulation result from Foster et al. [19] is shown in Figure 2 .

Based on the result, the holes and electrons’ densities position are almost similar to the reference as the condition referred to by Foster et al. [19] is in a steady state condition. The behaviour of the charge densities in Figure 2 is related to the dynamic mechanism of the perovskite solar cell. At the blend phase layer (perovskite), the absorption of light occurs and excites the electron-hole pair, resulting in the movement of holes and electrons [20] [21] . The electric field produced between HTL and ETL causes the electrons and holes to drift toward the selective layers.

At $x=250\text{nm}$, the value of electron density $\left(250\text{nm},t\right)$, becomes 3.9281 × 10²³ m⁻³ representing the behaviour of photo-generated electrons diffused from the blend phase layer into the ETL. The hole density at $x=250\text{nm}$ has a value of 2.055 × 10²⁴ m⁻³, showing that the accumulation occurs due to the selective charge properties at the interface of the blend phase (perovskite) layer to the ETL [2] . The holes accumulated are then blocked from entering ETL.

At $x=-250\text{nm}$, the value of hole density, $p\left(-250\text{nm},t\right)=5.3133\times {10}^{23}{\text{m}}^{-3}$, which is lower compared to the density of electrons. This situation represents a movement of the photo-generated holes from the blend phase (perovskite) layer into HTL. Meanwhile, the value of electron density, $n\left(-250\text{nm},t\right)$ piled up to 1.264 × 10²⁴ m⁻³ due to the selective layer properties at the interface of the HTL, thus blocking electrons from entering HTL [15] . The result shows that the concentration of charge density is important and can affect the performance of the solar cell. The effect of the parameter variation, such as thickness and the recombination coefficient, is discussed in the next section.

We would like to express our sincere appreciation toward Universiti Teknikal Malaysia Melaka (UTeM) and Center for Robotics and Industrial Automation (CeRIA) for the research support.