HelioSim: Drift-Diffusion Solar Cell Simulator for Perovskite Devices

A comprehensive MATLAB toolkit for simulating the physics and performance of perovskite solar cells using the drift-diffusion model with Chebfun spectral methods.

Table of Contents

1. Code Structure
2. Mathematical Model
3. Numerical Methods
4. Example: Perovskite Solar Cell
5. Installation and Usage
6. Advanced Features

Code Structure

HelioSim is organized into several core modules that work together to simulate solar cell behavior:

Core Components

1. SolarCellParamsOptimized.m

   • Contains physical constants, material parameters, and grid settings
   • Specialized for perovskite solar cells
   • Provides convenient parameter setting methods

2. DDSolverChebfunOptimized.m

   • Drift-diffusion equation solver using Chebfun spectral methods
   • Implements high-precision solutions for Poisson and continuity equations
   • Integrates interface handling and Scharfetter-Gummel discretization
   • Uses adaptive time stepping and Newton iteration

3. RecombinationModelsOptimized.m

   • Implements SRH, Auger, and radiative recombination models
   • Special handling for interface recombination
   • Supports energy-dependent trap models

4. OpticalGenerationOptimized.m

   • Carrier generation model using Beer-Lambert absorption
   • Supports wavelength-dependent absorption coefficients
   • Optimized for perovskite materials

5. JVAnalyzerOptimized.m

   • J-V curve analyzer
   • Calculates Voc, Jsc, fill factor, and efficiency
   • Implements maximum power point tracking
   • Optimized voltage scanning algorithm

6. VisualizerOptimized.m

   • Results visualization tool
   • Band diagram plotting
   • Carrier density visualization
   • J-V curve plotting
   • Electric field distribution visualization

7. main_perovskite_cell.m

   • Main script for perovskite solar cell simulation
   • Sets parameters and configuration
   • Runs simulation
   • Analyzes and visualizes results

Data Flow

1. main_perovskite_cell.m creates a SolarCellParamsOptimized instance and sets parameters
2. Creates a DDSolverChebfunOptimized instance with parameters and configuration
3. Solver internally creates RecombinationModelsOptimized and OpticalGenerationOptimized instances
4. Solver solves drift-diffusion equations and returns results
5. Creates JVAnalyzerOptimized instance to analyze performance
6. Creates VisualizerOptimized instance to visualize results

Mathematical Model

HelioSim implements a comprehensive drift-diffusion model that solves the coupled semiconductor equations:

Core Equations

1. Poisson's Equation:

   ∇²φ = q/ε₀ε * (p - n + N⁺ - N⁻)
   

   where φ is the electrostatic potential, q is the elementary charge, ε is the relative permittivity, n and p are electron and hole concentrations, and N⁺/N⁻ are ionized donor/acceptor concentrations.

2. Carrier Continuity Equations:

   ∂n/∂t = ∇·(Dn∇n + μn·n·∇φ) + G - R
   ∂p/∂t = ∇·(Dp∇p - μp·p·∇φ) + G - R
   

   where D is the diffusion coefficient, μ is the carrier mobility, G is the generation rate, and R is the recombination rate.

3. Current Densities:

   Jn = q·μn·n·E + q·Dn·∇n
   Jp = q·μp·p·E - q·Dp·∇p
   

   where E is the electric field.

Boundary Conditions

• Ohmic Contacts: Fixed carrier concentrations at the electrodes
• Interface Conditions: Continuity of electric displacement and quasi-Fermi levels at material interfaces

Physical Processes

The simulator includes detailed models for:

• Optical Generation: Beer-Lambert absorption model with AM1.5G spectrum
• Recombination Mechanisms:
  • Shockley-Read-Hall (SRH) recombination
  • Radiative (band-to-band) recombination
  • Auger recombination
  • Interface recombination
• Interface Physics: Band offsets and interface recombination
• Band Diagram Calculation: Including band bending at interfaces

Numerical Methods

HelioSim employs advanced numerical techniques for accurate and efficient simulation:

Spatial Discretization

• Chebfun Spectral Methods: Replaces finite differences with spectral methods
• Adaptive Grid Refinement: At interfaces for better resolution
• High-Precision Derivative Calculation: For accurate field and current calculations

Time Integration

• Implicit Time Stepping: For numerical stability
• Adaptive Time Step Control: Based on solution dynamics
• Newton Method: For solving nonlinear equation systems

Interface Handling

• Scharfetter-Gummel Discretization: For accurate current calculation
• Band Discontinuity Treatment: Accounts for energy band offsets
• Thermionic Emission Model: For carrier transport across interfaces

Recombination Models

• Position-Dependent Parameters: For different material regions
• Interface-Specific Recombination: Enhanced recombination at interfaces
• Trap Energy Distribution: For realistic defect modeling

Example: Perovskite Solar Cell

Device Structure

The default simulation models a typical perovskite solar cell with the following structure:

• ETL: TiO2 (100 nm)
• Absorber: MAPbI3 (500 nm)
• HTL: Spiro-OMeTAD (100 nm)

Key Parameters

TiO2 (ETL)

• Bandgap: 3.2 eV
• Electron affinity: 4.0 eV
• Dielectric constant: 9.0
• Electron mobility: 100 cm²/Vs
• Donor doping: 1×10¹⁷ cm⁻³

MAPbI3 (Absorber)

• Bandgap: 1.55 eV
• Electron affinity: 3.9 eV
• Dielectric constant: 25.0
• Electron/hole mobility: 20 cm²/Vs
• Intrinsic (undoped)
• Carrier lifetime: 1 μs

Spiro-OMeTAD (HTL)

• Bandgap: 3.0 eV
• Electron affinity: 2.1 eV
• Dielectric constant: 3.0
• Hole mobility: 50 cm²/Vs
• Acceptor doping: 1×10¹⁷ cm⁻³

Interface Parameters

• ETL/Absorber interface recombination velocity: 10⁴ cm/s
• Absorber/HTL interface recombination velocity: 10⁴ cm/s

Simulation Results

The simulation produces the following key performance metrics for the perovskite solar cell:

• Open-circuit voltage (Voc): ~1.0-1.1 V
• Short-circuit current density (Jsc): ~22-24 mA/cm²
• Fill factor (FF): ~0.75-0.80
• Power conversion efficiency (PCE): ~18-22%

The simulation also generates detailed visualizations including:

• Band diagram showing band bending at interfaces
• Carrier concentration profiles
• Electric field distribution
• J-V characteristic curve
• Recombination rate profiles

Installation and Usage

Requirements

• MATLAB (version 2016b or newer recommended)
• Chebfun library (included in the package)

Installation

1. Clone this repository or download all files
2. Open MATLAB
3. Navigate to the HelioSim directory
4. Run the main script to start the simulation:

   main_perovskite_cell

Basic Usage

% Run the main perovskite cell simulation
main_perovskite_cell

% To modify parameters before running:
params = SolarCellParamsOptimized();
params.L_absorber = 500e-7;  % 500 nm absorber thickness
params.Eg_abs = 1.55;        % 1.55 eV bandgap
params.mu_n_abs = 20;        % 20 cm²/Vs electron mobility
params.mu_p_abs = 20;        % 20 cm²/Vs hole mobility

% Configure simulation
config.t_max = 1e-9;          % Maximum simulation time
config.illumination = true;   % Enable illumination

% Create solver and run
solver = DDSolverChebfunOptimized(params, config);
results = solver.solve();

% Analyze J-V characteristics
analyzer = JVAnalyzerOptimized(params, solver);
jv_results = analyzer.generateJVCurve(-0.1, 1.2, 30);

% Visualize results
visualizer = VisualizerOptimized(params);
visualizer.setResults(results);
visualizer.plotBandDiagram();
visualizer.setJVResults(jv_results);
visualizer.plotJVCurve();

Advanced Features

Parameter Sweeps

% Example: Sweep absorber thickness
thicknesses = [300, 400, 500, 600, 700] * 1e-7;  % nm to cm
PCE = zeros(size(thicknesses));

for i = 1:length(thicknesses)
    params.L_absorber = thicknesses(i);
    % Update grid
    params.generateGrid();
    % Run simulation and get J-V results
    jv_results = analyzer.generateJVCurve();
    PCE(i) = jv_results.PCE;
    fprintf('Thickness = %.0f nm, PCE = %.2f%%\n', thicknesses(i)*1e7, PCE(i));
end

% Plot results
figure;
plot(thicknesses*1e7, PCE, 'o-', 'LineWidth', 2);
xlabel('Absorber Thickness (nm)');
ylabel('Power Conversion Efficiency (%)');
title('PCE vs. Absorber Thickness');
grid on;

Hysteresis Analysis

The simulator can model hysteresis effects in perovskite solar cells by performing forward and reverse voltage scans:

% Configure hysteresis analysis
config.scan_rate = 100;  % V/s
config.preconditioning = true;

% Run forward and reverse scans
[forward_results, reverse_results] = analyzer.performHysteresisAnalysis();

% Calculate hysteresis index
HI = analyzer.calculateHysteresisIndex(forward_results, reverse_results);
fprintf('Hysteresis Index: %.4f\n', HI);

% Visualize hysteresis
visualizer.plotHysteresisJVCurve(forward_results, reverse_results);

Custom Optical Generation

% Create custom generation profile
optical_gen = OpticalGenerationOptimized(params);
optical_gen.enable_interference = true;  % Enable interference effects
custom_gen = optical_gen.calculateGeneration();

% Use in simulation
solver.setGenerationProfile(custom_gen);

Interface Engineering

% Modify interface properties
params.S_ETL_abs = 1e3;  % ETL/absorber interface recombination velocity (cm/s)
params.S_abs_HTL = 1e3;  % absorber/HTL interface recombination velocity (cm/s)

% Add interface dipole
params.dipole_ETL_abs = 0.1;  % 0.1 eV dipole at ETL/absorber interface
recomb.B_rad_abs = 5e-10;        % Stronger radiative recombination
recomb.Cn_auger_abs = 1e-29;     % Faster Auger recombination
recomb.Cp_auger_abs = 1e-29;

Contributing

Contributions to HelioSim are welcome! Please feel free to submit pull requests or create issues for bugs and feature requests.

License

This project is licensed under the MIT License - see the LICENSE file for details.

Acknowledgments

• The drift-diffusion model implementation is based on semiconductor physics principles described in Semiconductor Device Physics and Design by Umesh K. Mishra and Jasprit Singh
• The solar cell architecture is inspired by typical perovskite and thin-film solar cell designs

Contact

For questions and support, please open an issue on the GitHub repository page.