SOLEY - A simulation package for solar cells

SOLEY User Manual Version 1.7 Zacharie Jehl Li-Kao December 10, 2025 Contents 1 Introduction 5 1.1 Purpose and Scope of SOLEY . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2 Key Features and Capabilities . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 What’s New in Version 1.7 . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.4 Underlying Physical Models (Brief Overview) . . . . . . . . . . . . . . . 6 1.5 Existing Solar Cell Modelling Software . . . . . . . . . . . . . . . . . . . 7 1.6 JustificationforSOLEY ........................... 8 1.6.1 High Accuracy and Predictive Power . . . . . . . . . . . . . . . . 8 1.6.2 Versatility and Adaptability . . . . . . . . . . . . . . . . . . . . . 8 1.6.3 Target Audience: A Tool for Experimentalists and Theorists . . . 9 1.6.4 A Powerful Complement to Traditional Tools . . . . . . . . . . . 9 1.7 PhilosophyofSOLEY ............................ 9 1.7.1 Core Principles of SOLEY . . . . . . . . . . . . . . . . . . . . . . 10 1.7.2 ScopeofSOLEY ........................... 10 1.7.3 What SOLEY Does Not Aim to Do . . . . . . . . . . . . . . . . . 11 2 Getting Started: A Basic Simulation Workflow 11 2.1 Launching the SOLEY Application . . . . . . . . . . . . . . . . . . . . . 11 2.2 Building a Simple Device Stack . . . . . . . . . . . . . . . . . . . . . . . 11 2.3 Defining Layer Properties . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.4 Setting Basic Calculation Parameters . . . . . . . . . . . . . . . . . . . . 12 2.5 Running the Optical Calculation . . . . . . . . . . . . . . . . . . . . . . 12 2.6 Interpreting the Optical Results Plot . . . . . . . . . . . . . . . . . . . . 12 2.7 Defining Absorber Layers and Bandgaps . . . . . . . . . . . . . . . . . . 12 2.8 Setting Basic Device Parameters . . . . . . . . . . . . . . . . . . . . . . . 13 2.9 Running the Device Calculation . . . . . . . . . . . . . . . . . . . . . . . 13 2.10 Interpreting the Device Results Table . . . . . . . . . . . . . . . . . . . . 13 2.11 Saving the Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3 Graphical User Interface (GUI) Overview 14 3.1 MainWindowLayout............................. 14 3.2 Layer Stack Management Panel (Top Left) . . . . . . . . . . . . . . . . . 14 3.3 Tabbed Controls Panel (Top Right) . . . . . . . . . . . . . . . . . . . . . 14 3.3.1 OpticalTab.............................. 15 3.3.2 Scattering & Coherence (Optical Settings) . . . . . . . . . . . . . 15 1 3.3.3 DeviceModelTab .......................... 15 3.3.4 AugerTab............................... 16 3.3.5 ResistanceTab ............................ 16 3.3.6 Collection Tab (New in v1.7) . . . . . . . . . . . . . . . . . . . . 16 3.3.7 DefectsTab.............................. 16 3.4 Optical Profile Plot (Bottom Left) . . . . . . . . . . . . . . . . . . . . . . 16 3.5 Device Performance Panel (Bottom Right) . . . . . . . . . . . . . . . . . 17 3.6 Buttons and Menu Actions (Summary) . . . . . . . . . . . . . . . . . . . 17 4 Methodology and Theoretical Background 17 4.1 Optical Model: Transfer Matrix Method (TMM) . . . . . . . . . . . . . . 17 4.1.1 CoreEquations ............................ 17 4.1.2 Handling of Coherent/Incoherent Layers . . . . . . . . . . . . . . 18 4.1.3 Assumptions.............................. 18 4.1.4 Advanced Optical Scattering and Partial Coherence (v1.7) . . . . 18 4.2 Layer Mixing: Bruggeman Effective Medium Approximation . . . . . . . 19 4.2.1 Governing Equation . . . . . . . . . . . . . . . . . . . . . . . . . 19 4.2.2 Assumptions.............................. 20 4.3 Electrical Model: Detailed Balance + Recombination . . . . . . . . . . . 20 4.3.1 Core Equation (Diode Equation Basis) . . . . . . . . . . . . . . . 20 4.3.2 Calculation of Jsc . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 4.3.3 CalculationofVoc .......................... 21 4.3.4 Recombination Mechanisms . . . . . . . . . . . . . . . . . . . . . 21 4.3.5 Voltage-Dependent Collection (Hecht Model) . . . . . . . . . . . . 22 4.3.6 Inclusion of Series and Shunt Resistance . . . . . . . . . . . . . . 22 4.3.7 Multijunction Simulation (2-Terminal) . . . . . . . . . . . . . . . 22 4.4 Limitations and Hypotheses . . . . . . . . . . . . . . . . . . . . . . . . . 22 5 Optical Simulation Module Details 23 5.1 TMM Implementation Specifics . . . . . . . . . . . . . . . . . . . . . . . 23 5.2 Defining Optical Constants (n, k file format) . . . . . . . . . . . . . . . . 23 5.3 Calculating Reflectance, Transmittance, and Absorbance . . . . . . . . . 24 5.4 Calculating Generation Profiles . . . . . . . . . . . . . . . . . . . . . . . 24 5.4.1 Direct Illumination . . . . . . . . . . . . . . . . . . . . . . . . . . 25 5.4.2 Diffuse Illumination . . . . . . . . . . . . . . . . . . . . . . . . . . 25 5.5 Handling Polarisation and Angle of Incidence . . . . . . . . . . . . . . . 25 5.6 Data Smoothing (Savitzky-Golay Filter) . . . . . . . . . . . . . . . . . . 25 5.7 Exporting Optical Profile and Generation Data . . . . . . . . . . . . . . 26 6 Electrical Device Modelling Module Details 26 6.1 Detailed Balance Model Implementation Specifics . . . . . . . . . . . . . 26 6.2 Calculation of Jsc, Voc, FF, and Efficiency . . . . . . . . . . . . . . . . . 26 6.3 Recombination Mechanisms In-Depth . . . . . . . . . . . . . . . . . . . . 27 6.3.1 Radiative Recombination Parameters . . . . . . . . . . . . . . . . 27 6.3.2 Auger Recombination Parameters . . . . . . . . . . . . . . . . . . 27 6.3.3 SRH Recombination (Defect Parameter Window Details) . . . . . 27 6.4 Calculation of J-V Curves (Light and Dark) . . . . . . . . . . . . . . . . 28 6.5 Multijunction Device Simulation Logic . . . . . . . . . . . . . . . . . . . 28 2 7 Multijunction Solar Cell J-V Calculation Methodology 29 7.1 PhysicalPrinciples .............................. 29 7.2 Computational Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 7.3 Mathematical Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . 29 7.4 Fill Factor Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 7.5 Advantages .................................. 30 7.6 Limitations .................................. 30 8 Luminescence Spectroscopy Module (NEW in v1.5) 31 8.1 Overview.................................... 31 8.2 AccessingtheModule............................. 31 8.3 InterfaceLayout................................ 31 8.4 Photoluminescence (PL) Parameters . . . . . . . . . . . . . . . . . . . . 31 8.4.1 BasicPLSettings........................... 31 8.4.2 PL Physical Model . . . . . . . . . . . . . . . . . . . . . . . . . . 31 8.5 Electroluminescence (EL) Parameters . . . . . . . . . . . . . . . . . . . . 32 8.5.1 BasicELSettings........................... 32 8.5.2 EL Physical Model . . . . . . . . . . . . . . . . . . . . . . . . . . 32 8.6 AdvancedOptions .............................. 32 8.6.1 Absorptivity Model Selection . . . . . . . . . . . . . . . . . . . . 32 8.6.2 Multi-Absorber Support . . . . . . . . . . . . . . . . . . . . . . . 32 8.6.3 PlotOptions ............................. 32 8.7 ResultsDisplay ................................ 33 8.8 ExportingData................................ 33 8.8.1 ExportData(CSV).......................... 33 8.8.2 ExportPlot.............................. 33 8.9 TypicalWorkflow............................... 33 8.9.1 Basic PL Measurement Simulation . . . . . . . . . . . . . . . . . 33 8.9.2 Voltage-Dependent PL . . . . . . . . . . . . . . . . . . . . . . . . 34 8.9.3 EL Characterisation . . . . . . . . . . . . . . . . . . . . . . . . . 34 8.9.4 Combined PL and EL Analysis . . . . . . . . . . . . . . . . . . . 34 8.10 Physical Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 8.10.1PLPeakPosition........................... 34 8.10.2PLIntensity.............................. 34 8.10.3 EL Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . 35 8.11Troubleshooting................................ 35 8.11.1NoPL/ELSignal........................... 35 8.11.2 Unrealistic Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . 35 8.11.3IntensityIssues ............................ 35 9 Shockley-Queisser Limit Calculator (NEW in v1.5) 35 9.1 Overview.................................... 35 9.2 Accessing the Calculator . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 9.3 InterfaceLayout................................ 36 9.4 SpectrumSelection .............................. 36 9.5 BasicParameters ............................... 36 9.5.1 BandgapRange............................ 36 9.5.2 Operating Conditions . . . . . . . . . . . . . . . . . . . . . . . . . 36 3 9.6 Parasitic Resistances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 9.6.1 Series Resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 9.6.2 ShuntResistance ........................... 37 9.7 Auger Recombination (Optional) . . . . . . . . . . . . . . . . . . . . . . 37 9.7.1 Enable/Disable Auger . . . . . . . . . . . . . . . . . . . . . . . . 37 9.7.2 AugerParameters........................... 37 9.8 Calculation and Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 9.8.1 Running Calculations . . . . . . . . . . . . . . . . . . . . . . . . . 38 9.8.2 ResultsPlot.............................. 38 9.8.3 ResultsTable ............................. 38 9.9 ExportingResults............................... 38 9.9.1 ExportPlot.............................. 38 9.9.2 ExportData.............................. 39 9.10TypicalUseCases............................... 39 9.10.1 Material Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 9.10.2 Loss Mechanism Analysis . . . . . . . . . . . . . . . . . . . . . . 39 9.10.3 Concentration Effects . . . . . . . . . . . . . . . . . . . . . . . . . 40 9.10.4 Multijunction Design . . . . . . . . . . . . . . . . . . . . . . . . . 40 9.11 Understanding the Physics . . . . . . . . . . . . . . . . . . . . . . . . . . 40 9.11.1 The Shockley-Queisser Limit . . . . . . . . . . . . . . . . . . . . . 40 9.11.2 Effect of Auger Recombination . . . . . . . . . . . . . . . . . . . 40 9.11.3 Impact of Resistances . . . . . . . . . . . . . . . . . . . . . . . . . 41 9.12AdvancedFeatures .............................. 41 9.12.1 Custom Spectrum Analysis . . . . . . . . . . . . . . . . . . . . . 41 9.12.2 Temperature Studies . . . . . . . . . . . . . . . . . . . . . . . . . 41 9.12.3 Material Comparison . . . . . . . . . . . . . . . . . . . . . . . . . 41 9.13 Limitations and Considerations . . . . . . . . . . . . . . . . . . . . . . . 42 10 Diode Analysis Tool (NEW in v1.7) 42 10.1Overview.................................... 42 10.2PhysicalQuantities.............................. 42 10.2.1 Local Ideality Factor (nid) ...................... 42 10.2.2 Differential Resistance (Rdiff ).................... 42 10.3InterfaceFeatures............................... 43 11 Advanced Features 43 11.1 Layer Thickness Optimisation . . . . . . . . . . . . . . . . . . . . . . . . 43 11.2BatchCalculations .............................. 44 11.3 Saving and Loading Simulation States . . . . . . . . . . . . . . . . . . . 44 11.4 Simulating Diffuse Light Effects . . . . . . . . . . . . . . . . . . . . . . . 45 12 File Formats 45 12.1 Optical Constant Files (.txt) ........................ 45 12.2 Spectrum Files (.txt) ............................ 46 12.3 Simulation State Files (.soley JSON structure) . . . . . . . . . . . . . . 46 12.4 Exported CSV File Formats . . . . . . . . . . . . . . . . . . . . . . . . . 47 4 13 Troubleshooting and FAQ 47 13.1 Common Errors and Solutions . . . . . . . . . . . . . . . . . . . . . . . . 47 13.2 Tips for Accurate Simulations . . . . . . . . . . . . . . . . . . . . . . . . 48 13.3 Frequently Asked Questions . . . . . . . . . . . . . . . . . . . . . . . . . 49 14 Version History 49 14.1 Version 1.7 (Current) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 14.2Version1.5................................... 50 14.3Version1.0.1.................................. 50 15 Appendix 50 15.1 List of Physical Constants Used . . . . . . . . . . . . . . . . . . . . . . . 50 15.2Acknowledgments............................... 50 15.3ContactandSupport............................. 51 15.4CitingSOLEY................................. 51 15.5 License and Terms of Use . . . . . . . . . . . . . . . . . . . . . . . . . . 51 1 Introduction 1.1 Purpose and Scope of SOLEY SOLEY (Solar Cell Optics and Electrics) is a simulation package designed for researchers, engineers, and students working with photovoltaic devices. Its primary purpose is to provide an efficient and accessible platform for modelling the optical and electrical performance of single-junction and multijunction solar cells. The software bridges the gap between complex drift-diffusion solvers and simpler analytical models, and it aims to offer a balance of accuracy, speed, and ease of use. SOLEY is intended for simulating planar, multilayer solar cell structures and predicting key performance metrics like Jsc,Voc, Fill Factor (FF), and efficiency (η). 1.2 Key Features and Capabilities SOLEY offers a range of features for comprehensive solar cell analysis: •Optical Simulation: Utilises the Transfer Matrix Method (TMM) to calculate reflectance, transmittance, absorbance, and optical generation profiles within the device stack. Supports both direct and diffuse illumination. •Electrical Device Modelling: Employs a detailed balance framework incorporating radiative, Auger, and Shockley-Read-Hall (SRH) recombination mechanisms to determine J-Vcharacteristics and performance parameters. Allows for detailed defect analysis via the SRH module which is designed to accommodate both experimentalists and theoreticians. •Luminescence Spectroscopy: NEW in v1.5 - Integrated photoluminescence (PL) and electroluminescence (EL) modelling using complete SOLEY physics for characterisation and analysis. 5 •Shockley-Queisser Limit Calculator: NEW in v1.5 - Standalone module for detailed balance analysis with customisable parameters including Auger recombination, series and shunt resistances. •Layer Mixing: Includes the Bruggeman Effective Medium Approximation (EMA) to simulate layers composed of mixed materials. •Optimisation: Features a tool for optimising the thickness of selected layers to maximise device efficiency based on a grid search algorithm. •Batch Processing: Enables automated parameter sweeps for efficient exploration of the device design space. Users can vary general parameters (like temperature) or absorber-specific parameters (like resistances or recombination coefficients). 1.3 What’s New in Version 1.7 This release introduces advanced physical modelling capabilities and characterisation tools: •Advanced Optical Scattering: Implementation of hybrid interface scattering, roughness-induced haze, and partial coherence modelling. •Diode Analysis Tool: A dedicated module for extracting local ideality factors (nid) and differential resistance (Rdiff ) from J-V curves. •Voltage-Dependent Collection: Implementation of the Hecht equation to model carrier collection losses limited by mobility-lifetime product (µτ). •Luminescence Spectroscopy Module: (Introduced v1.5) Full implementation of PL and EL calculations. •Standalone SQ Limit Calculator: (Introduced v1.5) Theoretical efficiency limit analysis. 1.4 Underlying Physical Models (Brief Overview) SOLEY’s calculations are based on established physical models: •Optics: The Transfer Matrix Method (TMM) calculates how light propagates through and is absorbed by the different layers of the solar cell. •Electronics: A detailed balance model, extending the Shockley-Queisser limit, is used. It considers the generation of carriers from absorbed light and their loss through various recombination pathways (radiative, Auger, SRH). The model incorporates the effects of series and shunt resistance. For multijunction cells, current matching between subcells is enforced for 2-terminal configurations. •Luminescence: Planck emission equations modified by device physics (quasiFermi level splitting, voltage-dependent recombination) to calculate realistic PL and EL spectra. These models allow for rapid yet physically meaningful simulations of device behaviour. A more detailed theoretical background is provided in Section 4. 6 1.5 Existing Solar Cell Modelling Software Numerous software tools have been developed to simulate and analyse photovoltaic (PV) devices, each adopting different theoretical and numerical approaches. Widely adopted tools include SCAPS, SolCore, AFORS-HET, WX-AMPS, OghmaNano, Sentaurus, and Silvaco. These programs typically employ advanced drift-diffusion models coupled with Poisson’s equation and carrier continuity equations. Specifically, drift-diffusion modelling involves solving the coupled semiconductor equations: Poisson equation: ∇ · (ϵ∇ψ) = −q(p−n+N+ D−N− A) (1) Electron continuity: 1 q∇ · Jn=G−Rn−∂n ∂t (2) Hole continuity: −1 q∇ · Jp=G−Rp−∂p ∂t (3) Electron drift-diffusion current: Jn=qµnnE +qDn∇n(4) Hole continuity current: Jp=qµppE −qDp∇p(5) where Jnand Jpare electron and hole current densities, Gand Rrepresent generation and recombination rates, N+ Dand N− Aare ionised donor and acceptor concentrations, and µn,µp,Dn, and Dpare carrier mobilities and diffusivities. Solving these coupled differential equations allows detailed modelling of carrier transport and recombination in semiconductor devices. The contribution of those packages for advancing the community’s understanding of solar cells cannot be overstated, and researchers in the field of PV are strongly encouraged to integrate one of those in their workflow. Strengths of existing software: •High fidelity modelling of semiconductor physics through the drift-diffusion approach. •Inclusion of comprehensive recombination mechanisms (Shockley-Read-Hall, radiative, Auger) and detailed material properties. •Capability to generate detailed electrical characteristics such as band diagrams, electric fields, and carrier profiles. •Established validation through extensive literature use. Hence, making a meaningful contribution in this landscape is challenging. After having used SCAPS extensively over the years, dating back to my days as a PhD student, and having occasionally played with some of the aforementioned software, I identified specific points which could be addressed by a new package: •Steep learning curve, requiring considerable training and experience. •Complex input parameter definitions, often requiring detailed knowledge of material properties for all layers in the stack. •High computational costs for complex structures, limiting rapid iteration. 7 •Difficulties with transparency and reproducibility of results due to intricate numerical configurations. Given these considerations, there could be a need within the PV research community for a complementary tool that balances accuracy, transparency, ease of use, and accessibility, motivating the development of SOLEY. 1.6 Justification for SOLEY Existing solar cell simulation tools provide highly detailed modelling approaches but often suffer from complexity, computational intensity, and a steep learning curve. SOLEY has been developed to provide a powerful, accurate, and flexible framework for photovoltaic device modelling that retains computational efficiency while maintaining predictive accuracy. Unlike many traditional simulation tools that require highly detailed input parameters and mesh-dependent numerical solvers, SOLEY adapts to the level of available information. It allows users to perform accurate device modelling whether they rely on experimentally extracted parameters from fitting or theoretical predictions from firstprinciples calculations. 1.6.1 High Accuracy and Predictive Power Despite its flexibility, SOLEY retains high predictive accuracy by incorporating fundamental physical models that govern photovoltaic operation: •Optical Modelling: Uses the Transfer Matrix Method (TMM) to precisely compute reflectance, absorbance, and transmittance in multilayer structures, ensuring accurate determination of light absorption profiles in complex architectures. •Charge Transport and Recombination: Implements a detailed balance framework while incorporating realistic recombination mechanisms, including ShockleyRead-Hall (SRH), Auger, and radiative recombination. This enables calculation of realistic open-circuit voltage, fill factor, and efficiency. •Numerical Stability: Employs computational techniques to prevent divergence of the TMM, ensuring energy conservation and physically meaningful results in a broad range of configurations (yes, it can still be broken if one tries hard enough). 1.6.2 Versatility and Adaptability One of SOLEY’s key strengths is its ability to seamlessly adapt to different levels of input complexity: •Experimentalists: Can directly use extracted parameters (e.g., recombination currents, defect densities, optical constants) from real devices to analyse performance and guide further optimisation. •DFT and First-Principles Modellers: Can incorporate parameters derived from first-principles calculations to predict solar cell performance before experimental realisation. 8 •Users with Limited Data: When detailed parameters are unavailable, SOLEY provides physically reasonable estimates to enable meaningful analysis, allowing users to progressively refine their models as more data become available. This flexibility ensures that SOLEY is not only a tool for rapid evaluation but can also be viewed as a comprehensive modelling package for more rigorous photovoltaic research. 1.6.3 Target Audience: A Tool for Experimentalists and Theorists SOLEY is designed to be accessible to a broad spectrum of researchers, providing modelling capabilities with well-reported equations without the complexity of traditional driftdiffusion solvers: •Experimental researchers: Enables precise interpretation of photovoltaic performance using extracted material parameters and device-specific measurements. •Density Functional Theory (DFT) and semiconductor theorists: Allows direct integration of computationally derived parameters to predict device efficiency and behaviour. •Educators and students: Serves as an intuitive platform to explore photovoltaic concepts with real-world physical modelling. 1.6.4 A Powerful Complement to Traditional Tools Rather than being a simplified version of existing tools, SOLEY is a powerful complement that excels in: •Accurate Performance Prediction: Provides results comparable to drift-diffusion solvers when applied within its methodological framework. •High Computational Efficiency: Avoids the overhead of finite-element meshing while delivering detailed device insights. •Broad Applicability: Supports single-junction, tandem, and multijunction devices with up to six absorbers, making it suitable for state-of-the-art solar cell architectures. By combining scientific rigour, computational efficiency, and user flexibility, SOLEY represents an unusual approach to solar cell modelling that can serve both the experimental and theoretical researchers. 1.7 Philosophy of SOLEY While SOLEY does have obvious limitations, like any tool, SOLEY is built on a physicsdriven, parameter-flexible approach that allows both high-precision modelling and quick exploratory analyses. This section defines the core principles of SOLEY, its scope, and what it does not aim to do. 9 •Temperature: Operating temperature in Kelvin. •Absorber Bandgaps: Input fields for the bandgap (eV) of each absorber (1-6). •Update Bandgaps Button: Applies manually entered bandgap values. 3.3.4 Auger Tab Input fields for Auger coefficients: •Electrons: Auger coefficients for electrons (e1-e6) in cm6/s •Holes: Auger coefficients for holes (h1-h6) in cm6/s 3.3.5 Resistance Tab Input fields for parasitic resistances: •Series Resistance: Rs1-Rs6 in Ohm·cm2 •Shunt Resistance: Rsh1-Rsh6 in Ohm·cm2 3.3.6 Collection Tab (New in v1.7) Controls for voltage-dependent carrier collection (Hecht Model): •Enable Collection Model: Checkbox to activate drift-length limited collection. •Mu-Tau (µτ): Mobility-lifetime product for each absorber (cm2/V). •Built-in Voltage (Vbi): The internal built-in potential (V) across the absorber. 3.3.7 Defects Tab Controls for SRH recombination modeling: •Absorber Buttons (1-6): Opens the Define Defect Profile window for each absorber. •Doping Type: Dropdown to select p-type or n-type doping for each absorber. 3.4 Optical Profile Plot (Bottom Left) Displays the results of the optical calculation: •Shows Reflectance (R), Transmittance (T), and individual layer Absorbance (A) versus wavelength. •Includes Total Absorbance and the sum R+T+A for verification. •Plot updates automatically after running the optical calculation. •Grid lines aid visualisation. 16 3.5 Device Performance Panel (Bottom Right) Controls electrical calculations and displays results: •Action Buttons: – CALCULATE DEVICE PARAMS: Main calculation button (blue, bold) – J-V Curves: Opens J-V plotting window (green) – Diode Analysis: Opens the diode analysis and extraction tool (orange) - NEW in v1.7 – Luminescence: Opens luminescence spectroscopy window (purple) - NEW in v1.5 – SQ Limit: Opens Shockley-Queisser limit calculator (brown) - NEW in v1.5 – Batch Calc: Opens batch calculation window (pink) – Optimise Stack: Opens layer optimisation window (cyan) – SQ Step: Calculate device with SQ radiative limit only (teal) – Clear Table: Clears the results table •Results Table: Displays calculated device parameters in a scrollable text area with fixed-width font for alignment. 3.6 Buttons and Menu Actions (Summary) Key action buttons are distributed across the panels as described above. There are no traditional File/Edit menus; all actions are controlled via buttons in the main interface. 4 Methodology and Theoretical Background This section describes the physical models and assumptions used within SOLEY. 4.1 Optical Model: Transfer Matrix Method (TMM) SOLEY calculates how light interacts with the solar cell stack using the Transfer Matrix Method (TMM). 4.1.1 Core Equations The TMM models each layer with a characteristic matrix. Light propagation across an interface between layer jand j+ 1 is described by an interface matrix Dj. Propagation through a layer jof thickness djis described by a propagation matrix Pj. For a stack of Nlayers, the total transfer matrix Mis the product of these individual matrices: M=D0·P1·D1·P2· · · · · PN·DN(6) where D0represents the interface from the incident medium (e.g., air) to the first layer, and DNrepresents the interface from the last layer to the substrate or exit medium. 17 From the elements of the total matrix M=M11 M12 M21 M22, the complex reflection coefficient rand transmission coefficient tare found: r=M21 M11 (7) t=1 M11 (8) Reflectance (R) and Transmittance (T) are then calculated as: R=|r|2(9) T= Re kz,N+1 kz,0|t|2(TE) or T= Re kz,N+1n2 0 kz,0n2 N+1 |t|2(TM) (10) where kzis the wavevector component perpendicular to the interfaces, and nis the refractive index of the incident (0) and exit (N+1) media. Absorbance (A) in the stack is calculated as A= 1 −R−T. Absorbance within a specific layer is found by calculating the Poynting vector difference at its boundaries. 4.1.2 Handling of Coherent/Incoherent Layers SOLEY implicitly assumes coherent light propagation throughout the stack, which is standard in TMM. For very thick layers where incoherence might occur (e.g., thick substrates), the TMM approach still provides a good approximation for overall R, T, and A, although it does not explicitly model phase scrambling. SOLEY includes numerical stability enhancements for thick layers. 4.1.3 Assumptions •Layers are optically homogeneous and isotropic. •Interfaces between layers are perfectly planar and parallel. •Light propagation is coherent within the entire stack. •Material optical constants (n, k) are known for the simulated wavelength range. 4.1.4 Advanced Optical Scattering and Partial Coherence (v1.7) SOLEY v1.7 introduces a sophisticated hybrid scattering model that bridges the gap between wave optics (TMM) and geometric ray optics. This allows for accurate simulation of textured interfaces, rough layers, and thick substrates where coherence is lost. Scalar Scattering Theory (Interface Roughness) Roughness at an interface is modelled using the Scalar Scattering Theory (SST). The coherent (specular) transmittance and reflectance are attenuated by a ’Debye-Waller’ like factor S, determined by the RMS roughness σ: Scoh = exp "−2πσ∆n λ2#(11) where ∆n=|n1−n2|is the refractive index contrast at the interface. Light that is not specularly transmitted/reflected (1 −Scoh) is considered ”scattered” or ”haze”. 18 Hybrid Scattering and Flux Redistribution Unlike simple attenuation models, SOLEY tracks the scattered flux. This ”Haze” is redistributed into angular cones defined by either a Lambertian distribution (isotropic scattering) or a Gaussian distribution (centred at a characteristic angle). The fraction of scattered light that is reflected back from an interface (Rdiff ) vs. transmitted (Tdiff ) is calculated by integrating the Fresnel coefficients over the scattering angles weighted by the distribution function W(θ): Rdiff =Rπ/2 0RFresnel(θ)W(θ) sin(θ)dθ Rπ/2 0W(θ) sin(θ)dθ (12) For a Lambertian surface, W(θ) = cos(θ), leading to the sin(2θ) weighting used in the code. Geometric Light Trapping (Net-Radiation Method) For thick, rough absorbers, scattered light undergoes multiple internal reflections, enhancing the effective path length. SOLEY employs a net-radiation algebra approach to calculate this enhancement factor Fpath: Fpath =1 + Rdiff rear 1−Rdiff rear Rdiff front(Tdiff int )2(13) This factor scales the absorption within the layer, effectively modelling light trapping schemes found in textured silicon or rough thin-film devices. Partial Coherence Standard TMM assumes infinite coherence length, leading to interference fringes that may not exist in thick layers (e.g., glass substrates). SOLEY solves this by spectral averaging. The user defines a coherence length Lc. The resulting spectrum is the convolution of the coherent TMM spectrum with a Gaussian function whose width ∆λdepends on Lc: ∆λ≈λ2 neff Lc (14) This rigorously removes unphysical interference fringes while conserving energy. 4.2 Layer Mixing: Bruggeman Effective Medium Approximation When the ’Mix’ option is enabled for a layer, SOLEY uses the Bruggeman Effective Medium Approximation (EMA) to calculate the effective optical constants of the mixture between that layer and the one below it. 4.2.1 Governing Equation The Bruggeman EMA solves the following equation for the effective dielectric permittivity ϵeff of a two-component mixture: f1 ϵ1−ϵeff ϵ1+ 2ϵeff +f2 ϵ2−ϵeff ϵ2+ 2ϵeff = 0 (15) where f1, f2are the volume fractions of material 1 and 2 (f1+f2= 1), and ϵ1, ϵ2are their respective complex dielectric permittivities (ϵ= (n+ik)2). SOLEY solves this equation iteratively at each wavelength to find ϵeff , from which the effective neff and keff are derived. 19 4.2.2 Assumptions •The mixture is random and homogeneous on a macroscopic scale. •The inclusions (minority component) are roughly spherical. •The size of the inclusions is much smaller than the wavelength of light. 4.3 Electrical Model: Detailed Balance + Recombination SOLEY uses a detailed balance approach, similar to the Shockley-Queisser limit, but extended to include non-radiative recombination and resistive losses. 4.3.1 Core Equation (Diode Equation Basis) The current-voltage (J-V) characteristic of each subcell (absorber) is modelled using a diode equation: J(V) = Jph −Jrec(V)−Jshunt(V) (16) where Jph is the photogenerated current density (calculated from optical absorption), Jrec(V) is the voltage-dependent recombination current density, and Jshunt(V) represents current loss through shunt pathways. The terminal voltage is related to the internal junction voltage Vjby series resistance Rs:V=Vj−J·Rs. The recombination current is the sum of different mechanisms: Jrec(Vj) = J0,rad e qVj kT −1+J0,Auger e qVj kT −1+J0,SRH(Vj)e qVj nSRH kT −1(17) where J0terms are saturation current densities for radiative, Auger, and SRH recombination, kis Boltzmann’s constant, Tis temperature, qis elementary charge, and nSRH is the SRH ideality factor. Note that J0,SRH can be voltage-dependent if a custom equation is used, but the recombination paths by default do not depend on the voltage. The shunt current is modelled as Jshunt(V)=(V+J·Rs)/Rsh, where Rsh is the shunt resistance. SOLEY solves the implicit equation for J at each V numerically to obtain the J-V curve. 4.3.2 Calculation of Jsc The short-circuit current density Jsc is ideally equal to Jph.Jph is calculated by integrating the absorbed photon flux above the bandgap Egfor each absorber layer, using the generation profile from the TMM simulation: Jph =qZ∞ Eg Φabs(E)dE (18) where Φabs(E) is the absorbed photon flux per unit energy. SOLEY calculates this from the layer absorbance A(λ) and the incident spectrum Φin(λ): Jph =qZλmax λmin A(λ)Φin(λ)dλ (19) where the integral is performed over wavelengths corresponding to energies above Eg. The effective Jsc used in calculations accounts for early recombination onset and resistive losses near V=0. 20 4.3.3 Calculation of Voc The open-circuit voltage Voc is the voltage where the net current J(V) is zero. It is found by solving: Jph =Jrec(Voc) + Jshunt(Voc) (20) This equation is solved numerically by SOLEY. 4.3.4 Recombination Mechanisms •Radiative (J0 rad): Calculated based on detailed balance, assuming the cell emits blackbody radiation filtered by its absorptivity above the bandgap. The equation used is: J0,rad =1 f 2πq(kT)3 h3c2Z∞ xg x2 ex−1dx ≈1 f 2πq(kT)3 h3c2e−xg(21) where xg=Eg/kT,his Planck’s constant, cis the speed of light, and fis the ´etendue factor (geometry factor). •Auger (J0 auger): Represents three-particle recombination. The saturation current J0,Auger depends on the Auger coefficients (Cn, Cp), intrinsic carrier concentration (ni), doping concentration (Ndop), and layer thickness (d): J0,Auger ≈q·d·Cmaj ·N2 dop ·ni(22) where Cmaj is the coefficient for the majority carrier (Cpfor p-type, Cnfor n-type). SOLEY uses user-provided Cnand Cpvalues and includes temperature and bandgap dependencies for these coefficients. •Shockley-Read-Hall (SRH): Recombination via defect states within the bandgap. SOLEY offers two ways to define J0,SRH and its ideality factor nSRH: 1. Microscopic Parameters (Scaffidi Model): Calculates J0,SRH from fundamental defect parameters (trap density Nt, capture cross-section σ, thermal velocity vth, trap energy level Et, doping, built-in voltage Vbi, etc.) and interface recombination velocity (Sp). It calculates bulk (J00,bulk) and interface (J00,interface) prefactors and uses an activation energy (Ea) approach: J00,bulk ∝Nt·σ·vth ·qNcNv/Ndop (23) J00,interface ∝Sp·Ndop (24) J0,SRH = (J00,bulk +J00,interface) exp −Ea nSRHkT (25) Assumptions: Single dominant trap level, specific model for depletion width and interface recombination. This is mostly based on Cuevas’ work. 2. Custom Equation: Allows the user to define J0,SRH as an arbitrary function of voltage (V) and other material/device parameters using Python syntax. This provides maximum flexibility to implement custom or empirical recombination models. 21 4.3.5 Voltage-Dependent Collection (Hecht Model) In materials with low carrier mobility or short lifetimes (e.g., certain perovskites, organic PV, or degraded devices), not all photogenerated carriers reach the contacts. SOLEY v1.7 implements the Hecht equation to model this collection efficiency ηc(V): ηc(V) = Ld(V) d1−exp −d Ld(V) (26) where dis the absorber thickness and Ld(V) is the voltage-dependent drift length: Ld(V) = µτ Vbi −V d(27) Here, µτ is the mobility-lifetime product and Vbi is the built-in voltage. The photogenerated current becomes Jph(V) = Jopt ×ηc(V). 4.3.6 Inclusion of Series and Shunt Resistance Series resistance (Rs) reduces the output voltage and fill factor. Shunt resistance (Rsh) provides an alternative path for current, reducing Voc and fill factor. SOLEY includes these as lumped parameters in the diode equation, affecting the terminal voltage and current. 4.3.7 Multijunction Simulation (2-Terminal) For devices with multiple absorbers, SOLEY assumes a 2-terminal configuration. It calculates the J-V curve for each subcell individually. The overall device J-V curve is then determined by the principle of current matching: the total current density cannot exceed the lowest Jsc among the series-connected subcells. The total Voc is the sum of the individual subcell Voc values (assuming negligible recombination at tunnel junctions). The fill factor and efficiency are calculated for the current-limited multijunction device. 4.4 Limitations and Hypotheses Users should be aware of the inherent assumptions and limitations: •1D Simulation: SOLEY assumes infinite lateral extent and uniformity. It cannot model edge effects, localised defects, or non-uniform illumination. •Simplified Transport: The detailed balance model does not explicitly calculate carrier concentration profiles or electric fields within the device like drift-diffusion solvers do. Transport is implicitly handled via recombination parameters and ideal collection assumptions (modified by resistance). •SRH Model Assumptions: When using the Scaffidi model, it assumes a single, dominant trap level and specific forms for bulk and interface recombination. The custom J0approach relies entirely on the user’s input model. •Interface Modelling: Interfaces are treated primarily through their optical properties (TMM) and potentially via interface recombination terms (J00,interface or custom J0). Detailed interface physics (band alignment, interface dipoles, specific interface traps) are not explicitly modelled as in drift-diffusion. 22 •Thermal Effects: Temperature is a fixed input parameter. SOLEY does not calculate self-heating effects within the device during operation. •Coherence Assumption: TMM assumes coherent light propagation, which may not be fully accurate for very thick, non-specular layers. That’s why smoothing functionality and numerical stability enhancements were added. 5 Optical Simulation Module Details This section provides more detail on the optical calculations performed by SOLEY. 5.1 TMM Implementation Specifics SOLEY uses a standard recursive TMM algorithm. To handle potentially thick or highly absorbing layers where numerical overflow or underflow can occur in the propagation matrix calculation (e±ikzd), the implementation includes: •Scaling: If the imaginary part of the exponent kzdbecomes too large (e.g., ¿ 350), the exponent is scaled down before calculating the exponential, and a corresponding scaling factor is applied to the matrix elements to maintain correctness while avoiding extreme values. •Sublayering (Experimental): For very thick absorbing layers, the propagation matrix calculation can optionally be split into multiple sub-steps to improve stability, although this increases computation time. This is handled internally based on layer thickness and absorption coefficient. •Parallel Processing: Optical calculations (R, T, A) for each wavelength are independent. SOLEY utilises multiprocessing to distribute these calculations across available CPU cores, significantly speeding up the simulation, especially for wide wavelength ranges or fine steps. 5.2 Defining Optical Constants (n, k file format) Each layer requires a file containing its optical constants (refractive index nand extinction coefficient k). •Format: The file must be a plain text file (.txt). •Columns: It should contain three columns of numerical data separated by whitespace (spaces or tabs): 1. Wavelength (in nanometres, nm) 2. Refractive Index (n, unitless) 3. Extinction Coefficient (k, unitless) •Order: Wavelengths should preferably be sorted in ascending order. •Comments: Lines starting with ’%’ are ignored as comments. 23 •Location: It is recommended to keep these files in a subfolder named Optical Indices within the SOLEY application directory or the directory where simulation files are saved/loaded. SOLEY will search in multiple locations (relative to the simulation file, relative to the executable, standard folders) to find the required files. SOLEY uses linear interpolation to find nand kvalues at wavelengths required for the simulation that may fall between the points defined in the file. Extrapolation is used for wavelengths outside the defined range. 5.3 Calculating Reflectance, Transmittance, and Absorbance After running the optical calculation: •Reflectance (R): The fraction of incident light intensity reflected from the front surface of the device stack. •Transmittance (T): The fraction of incident light intensity transmitted through the entire device stack. •Absorbance (A): Calculated in two ways: – Total Stack Absorbance: Calculated as 1 −R−T. – Layer-Specific Absorbance: Calculated for layers where the ’Calc A’ checkbox is ticked. This is done by computing the difference in the Poynting vector (energy flow) at the front and back interfaces of that specific layer, normalised by the incident Poynting vector. This represents the fraction of incident light absorbed only within that layer. The results are plotted against wavelength in the optical profile plot. 5.4 Calculating Generation Profiles The optical generation rate G(x, λ) (photons generated per unit volume per unit time per unit wavelength) at a position xwithin a layer is proportional to the local light intensity I(x, λ) and the absorption coefficient α(λ) of the material at that wavelength: G(x, λ)∝α(λ)·I(x, λ) (28) SOLEY calculates the electric field profile E(x, λ) within each layer using the TMM results. The local intensity is I(x, λ)∝ |E(x, λ)|2. The absorption coefficient is α(λ) = 4πk(λ)/λ. The total generation rate Gtot(x) (photons/m3/s) at position xis obtained by integrating over the incident spectrum Φin(λ) (photons/m2/s/nm): Gtot(x) = Zλmax λmin α(λ)I(x, λ) Iin(λ)Φin(λ)dλ (29) where Iin(λ) is the incident intensity used for normalisation within the TMM calculation. SOLEY performs this calculation numerically for layers marked with ’Calc A’. 24 5.4.1 Direct Illumination Calculates the generation profile assuming a single angle of incidence specified by the user. 5.4.2 Diffuse Illumination Calculates the generation profile assuming light arrives equally from all directions in a hemisphere. This involves: •Calculating the intensity profile I(x, λ, θ) for multiple incidence angles θ. •Performing a weighted average over these angles, typically weighted by sin(2θ) to account for the solid angle and Lambertian distribution. •Integrating the angle-averaged generation rate over the incident spectrum. The number of angle steps used for the approximation is set in the GUI. 5.5 Handling Polarisation and Angle of Incidence •Polarisation: TMM calculations are performed separately for TE (s-polarisation) and TM (p-polarisation) components of light, as the interface Fresnel coefficients differ. For ’Unpolarised’ light, SOLEY calculates both TE and TM results and averages them: Resultunpol = (ResultT E +ResultT M )/2. At normal incidence (0 degrees), TE and TM results are identical, so only one calculation is needed. •Angle of Incidence: The angle θ(relative to the surface normal) affects the wavevector components (kx, kz) within each layer via Snell’s Law, which modifies the TMM matrices. SOLEY correctly accounts for this angle dependence. 5.6 Data Smoothing (Savitzky-Golay Filter) Optical interference effects can cause rapid oscillations in the calculated R, T, and A spectra when a layer is very thick. This is particularly problematic when using a glass substrate or superstrate, for example. •If ’Enable Curve Smoothing’ is checked, SOLEY applies a Savitzky-Golay filter to the calculated R, T, and A data after the TMM calculation. •This filter fits a polynomial to a window of data points, effectively smoothing out high-frequency noise or interference fringes while preserving broader spectral features. •The ’Window Length’ (must be odd) and ’Poly Order’ parameters control the degree of smoothing. Larger windows or lower orders result in more smoothing. •Smoothing is applied only for display and export; the underlying electrical calculations typically use the unsmoothed absorbance data for accuracy. 25 8.5 Electroluminescence (EL) Parameters 8.5.1 Basic EL Settings •Enable EL: Checkbox to include EL in calculations •Applied voltage (V): Forward bias voltage for injection •Internal efficiency : Quantum efficiency factor (0-1) 8.5.2 EL Physical Model The EL spectrum is calculated using: ΦEL(E) = ηint ·2π h3c2·E2·A(E)·1 e(E−qV )/kT −1(39) where: •Uses compute jv curve for absorber() for device physics at applied voltage •Accounts for series resistance, recombination, and injection effects •Internal efficiency factor scales the emission intensity 8.6 Advanced Options 8.6.1 Absorptivity Model Selection •Use step function: Simplified model assuming A(E) = 1 for E ¿ Eg, A(E) = 0 otherwise •Use detailed TMM: Full optical calculation with wavelength-dependent absorptivity and reabsorption effects (recommended for accurate spectra) 8.6.2 Multi-Absorber Support •Select which absorber layer to calculate luminescence for •Each absorber can have different PL and EL characteristics •Useful for studying individual subcells in multijunction devices 8.6.3 Plot Options •Y-axis Scale: Toggle between logarithmic (default) and linear scale •Dual Axes: Bottom axis shows photon energy (eV), top axis shows wavelength (nm) •Update Visibility: Show/hide PL or EL curves independently 32 8.7 Results Display The results textbox shows calculated parameters: •Quasi-Fermi level splitting (∆µ) in eV •Implied open-circuit voltage from splitting •Peak emission wavelength/energy •Integrated photon flux •Operating conditions (temperature, voltage, excitation power) 8.8 Exporting Data 8.8.1 Export Data (CSV) Creates a CSV file containing: •Energy (eV) and wavelength (nm) columns •PL photon flux (photons/m²/s/eV) if enabled •EL photon flux (photons/m²/s/eV) if enabled •Metadata header with calculation parameters 8.8.2 Export Plot Saves the current plot as a high-resolution image file (PNG, PDF, or SVG format). 8.9 Typical Workflow 8.9.1 Basic PL Measurement Simulation 1. Complete optical TMM simulation in main SOLEY window 2. Define absorber layers and bandgaps 3. Open Luminescence window 4. Set excitation wavelength (e.g., 532 nm for green laser) 5. Set excitation power (typical: 1-100 mW/cm²) 6. Enable ”Use detailed TMM absorptivity” for accuracy 7. Click ”Calculate Spectra” 8. Analyze PL peak position and shape 33 8.9.2 Voltage-Dependent PL 1. Follow steps 1-6 from basic PL 2. Switch to ”Fixed V” mode 3. Enter desired voltage 4. Calculate and observe how spectrum shifts with voltage 5. Repeat for multiple voltages to study recombination dynamics 8.9.3 EL Characterisation 1. Ensure device parameters are properly defined (resistances, recombination) 2. Open Luminescence window 3. Enable EL, disable PL 4. Set applied forward bias voltage 5. Adjust internal efficiency if known (default 1.0) 6. Calculate to see emission under electrical injection 8.9.4 Combined PL and EL Analysis 1. Enable both PL and EL 2. Set appropriate parameters for each 3. Calculate to compare photoand electro-generated emission 4. Use logarithmic scale to see both spectra if intensities differ greatly 5. Export data for quantitative comparison 8.10 Physical Interpretation 8.10.1 PL Peak Position •Typically near the bandgap energy •Red-shift indicates band-gap renormalisation or Stokes shift •Blue-shift possible with high excitation (band-filling effects) •Shape depends on absorptivity profile and reabsorption 8.10.2 PL Intensity •Higher intensity indicates better material quality (lower non-radiative recombination) •Absolute intensity depends on excitation power and extraction efficiency •Ratio of PL to excitation power gives external quantum efficiency information 34 8.10.3 EL Characteristics •EL onset voltage near open-circuit voltage •Peak position should match PL peak (same radiative transitions) •EL efficiency decreases with increasing non-radiative recombination •Series resistance causes voltage shift between injection and emission 8.11 Troubleshooting 8.11.1 No PL/EL Signal •Check that optical calculations completed successfully •Verify absorber layer is defined with correct bandgap •For PL: excitation wavelength must be above bandgap •For EL: applied voltage must be sufficient for injection •Check if absorptivity is being calculated (Calc A checkbox) 8.11.2 Unrealistic Spectra •Try ”detailed TMM absorptivity” instead of step function •Verify TMM wavelength range covers emission region •Check temperature setting (affects spectral shape) •For EL: verify device physics parameters are realistic 8.11.3 Intensity Issues •Use logarithmic scale for wide intensity ranges •For PL: higher excitation power increases signal •For EL: internal efficiency factor scales output •Absolute intensities depend on extraction geometry (not explicitly modeled) 9 Shockley-Queisser Limit Calculator (NEW in v1.5) 9.1 Overview The SQ Limit Calculator is a standalone tool for exploring theoretical efficiency limits of solar cells. It extends the classic Shockley-Queisser framework to include Auger recombination and parasitic resistances, allowing more realistic limit calculations. 35 9.2 Accessing the Calculator •Click the SQ Limit button in the Device Performance panel (brown button) •Opens as an independent window •Does not require optical calculations (uses direct spectral integration) 9.3 Interface Layout The SQ Limit window contains: •Top Section: Spectrum selection and loading •Left Panel: Parameter controls organized in collapsible sections •Right Panel: Results plot showing efficiency vs bandgap •Bottom Section: Results table with numerical values 9.4 Spectrum Selection •Load Spectrum: Select incident spectrum file (same format as main SOLEY) •Spectrum must be in wavelength-power format (nm, W/m²/nm) •Automatically converts to energy-power format internally •Displays spectrum filename in label after loading 9.5 Basic Parameters 9.5.1 Bandgap Range •E g min (eV): Minimum bandgap for sweep (default 0.5) •E g max (eV): Maximum bandgap for sweep (default 4.0) •E g step (eV): Step size for bandgap sweep (default 0.01) •Finer steps give smoother curves but take longer to calculate 9.5.2 Operating Conditions •Temperature (K): Cell operating temperature (default 300) •Injection Factor: Light concentration (logarithmic slider) –Range: 0.001 to 46300 suns –Default: 1.0 (standard 1-sun AM1.5G) –Displays equivalent power density in mW/cm² 36 9.6 Parasitic Resistances 9.6.1 Series Resistance •R s (·cm²): Lumped series resistance (default 0) •Reduces output voltage and fill factor •Typical values: 0.1-10 ·cm²for real devices •Logarithmic slider for wide range access 9.6.2 Shunt Resistance •R sh (·cm²): Lumped shunt resistance (default 10000) •Provides leakage current path •Typical values: 1000-100000 ·cm²for good devices •Logarithmic slider for wide range access 9.7 Auger Recombination (Optional) 9.7.1 Enable/Disable Auger •Checkbox to include Auger recombination in calculations •When disabled, uses purely radiative limit •When enabled, requires additional parameters 9.7.2 Auger Parameters •C A n (cm/s): Electron Auger coefficient •C A p (cm/s): Hole Auger coefficient •Thickness (m): Active layer thickness •m e eff (m): Effective electron mass •m h eff (m): Effective hole mass These parameters determine the Auger saturation current: J0,Auger =q·d·CA·N2 dop ·ni(40) 37 9.8 Calculation and Results 9.8.1 Running Calculations •Click Calculate SQ Limit button •Sweeps through specified bandgap range •For each bandgap: –Calculates ideal Jsc from spectrum –Calculates J0 rad from detailed balance –Includes J0 Auger if enabled –Solves for Voc, FF, and efficiency considering resistances •Progress shown in status messages 9.8.2 Results Plot Displays efficiency ( •Peak efficiency and corresponding optimal bandgap •Effect of non-radiative losses (if Auger enabled) •Impact of resistances on peak efficiency •Shape of efficiency curve for material selection 9.8.3 Results Table Shows detailed parameters at optimal bandgap: •Optimal E g (eV) •Maximum efficiency ( •Jsc (mA/cm²) at optimal gap •Voc (V) at optimal gap •Fill Factor ( •J0 rad and J0 Auger (if applicable) 9.9 Exporting Results 9.9.1 Export Plot Saves the efficiency vs bandgap plot as an image file. 38 9.9.2 Export Data Creates CSV file with columns: •Bandgap (eV) •Efficiency ( •Jsc (mA/cm²) •Voc (V) •FF ( •J0 rad (A/cm²) •J0 Auger (A/cm²) if enabled 9.10 Typical Use Cases 9.10.1 Material Selection 1. Load appropriate spectrum (e.g., AM1.5G for terrestrial) 2. Set realistic temperature (300K typical) 3. Keep resistances at zero for ideal limit 4. Disable Auger for fundamental Shockley-Queisser limit 5. Calculate to find optimal bandgap for given spectrum 6. Compare with candidate material bandgaps 9.10.2 Loss Mechanism Analysis 1. Start with ideal case (no Auger, no resistances) 2. Note peak efficiency 3. Enable Auger with material-specific coefficients 4. Observe efficiency reduction 5. Add realistic resistances 6. Quantify each loss mechanism’s impact 39 9.10.3 Concentration Effects 1. Set up with realistic Auger parameters 2. Calculate at 1 sun (default) 3. Increase injection factor (concentration) 4. Observe how Voc increases (logarithmically) 5. Note that Auger losses increase with concentration 6. Find optimal concentration considering both effects 9.10.4 Multijunction Design 1. Calculate single-junction limit for reference 2. For tandem: identify two peak regions 3. Top cell: higher bandgap (around first peak) 4. Bottom cell: lower bandgap (around second peak) 5. Sum of subcell efficiencies gives tandem limit estimate 6. Consider current-matching requirements 9.11 Understanding the Physics 9.11.1 The Shockley-Queisser Limit The fundamental efficiency limit arises from: •Thermalisation: Photons above Eg lose excess energy as heat •Sub-bandgap loss: Photons below Eg are not absorbed •Radiative recombination: Unavoidable in thermal equilibrium •Entropy: Conversion of high-energy photons to voltage-limited work 9.11.2 Effect of Auger Recombination •Three-particle process: scales as n3or p3 •Becomes dominant in narrow-gap materials •Particularly important at high concentrations •Reduces Voc by providing additional recombination path •More severe in highly doped materials 40 9.11.3 Impact of Resistances •Series resistance: Mainly reduces FF, slight Voc loss –Voltage drop: Vloss =J·Rs –Maximum at maximum power point –More severe at high current densities •Shunt resistance: Reduces Voc and FF –Leakage current: Jleak =V/Rsh –Most severe at low voltages –Limits Voc if too low 9.12 Advanced Features 9.12.1 Custom Spectrum Analysis •Load custom spectrum files for specific applications •Indoor lighting: different optimal bandgap than solar •Space: AM0 spectrum gives different limits •Monochromatic: narrow peak, very application-specific 9.12.2 Temperature Studies •Vary temperature to see thermal effects •Higher T: lower Voc (entropy loss) •Lower T: higher efficiency (reduced recombination) •Important for space applications or hot-climate deployment 9.12.3 Material Comparison •Calculate with different Auger coefficients •Compare direct vs indirect bandgap materials •Assess thickness requirements (for Auger calculation) •Evaluate different doping levels 41 – Solution: Add layers using the ’Add Layer’ button. Check the ’Absorber’ box for the relevant layer(s) in the Layer Stack panel. •Error during calculations (e.g., ValueError, LinAlgError, Brentq failure) – Cause: Often due to invalid input parameters (e.g., non-numeric values, zero thickness, extreme resistance values) or numerical instability in the TMM or device solvers. Can also occur with highly unusual optical constants or extremely thin/thick layers. – Solution: Double-check all input parameters in the GUI, especially thicknesses, resistances, Auger coefficients, and SRH parameters. Ensure optical constant files are valid. Try slightly adjusting layer thicknesses or the wavelength step. If using custom SRH equations, verify their numerical stability. Check console output for more specific error messages if running from source. •GUI unresponsive during long calculations (Optimisation, Batch) – Cause: Intensive calculations run in separate threads, but GUI updates might lag or be unresponsive when calculations are not multithreaded. – Solution: Be patient. Monitor the console to confirm that calculations are ongoing. •Luminescence window shows no signal – Cause: TMM calculations not completed, or absorber not properly defined. – Solution: Run optical calculations first, ensure absorber checkbox is ticked and bandgap is set. 13.2 Tips for Accurate Simulations •Accurate Optical Constants: Use high-quality, experimentally measured nand kdata covering the full wavelength range of interest. Ensure the file format is correct. •Sufficient Wavelength Resolution: Use a small enough ’Wavelength Step’ (e.g., 10 nm at least) to capture sharp spectral features or interference effects, especially for optical optimisation. •Realistic Recombination Parameters: Use experimentally relevant values or physically sound estimates for SRH, Auger, and radiative recombination parameters (J0, n-factors, coefficients). Use the Defect Parameter window for detailed SRH modelling if possible. •Reasonable Resistances: Input realistic values for series (Rs) and shunt (Rsh) resistance based on device architecture and expected quality. Very low Rsh or very high Rscan significantly impact results. •Verify R+T+A Sum: Check the optical plot. The R+T+A curve should be close to 1 across the spectrum, indicating energy conservation in the optical model. Deviations might suggest issues with optical constants or numerical stability. 48 •Start Simple: Begin with a basic model and gradually add complexity (e.g., detailed SRH, layer mixing) to understand the impact of each parameter. 13.3 Frequently Asked Questions •Q: Can SOLEY simulate textured surfaces? •A: Yes since version 1.7 =). And it’s pretty cool because it does take light trapping into account, not just scattered losses. Publication to follow soon. •Q: How accurate is the detailed balance model compared to drift-diffusion? •A: When recombination parameters are well-characterised, the detailed balance model in SOLEY can provide results (Voc, Jsc, FF, Eff) very close to drift-diffusion solvers, especially for devices not heavily limited by complex transport phenomena. However, it does not provide spatial information like band diagrams or carrier profiles. However, the new inclusion (1.7) of voltage dependent collection brings it really close to what is achievable with drift diffusion. •Q: Can I model organic solar cells (OSCs) or perovskites? •A: Yes or Perovskite, no for organic. And the degradation of Perovskite isn’t yet taken into account. Organic are nearly impossible to model rigorously, at least for my level of knowledge. •Q: Should I use the SQ Limit Calculator or the main SOLEY interface? •A: Use the SQ Limit Calculator for theoretical limit studies and material selection. Use the main SOLEY interface for realistic device modeling with actual layer stacks, optical effects, and detailed recombination mechanisms. •Q: Can I import optical data from ellipsometry directly? •A: SOLEY requires data in the simple three-column format (wavelength, n, k). Most ellipsometry software can export to this format. Ensure wavelengths are in nm and sorted in ascending order. 14 Version History 14.1 Version 1.7 (Current) •New Physics: Hybrid optical scattering/haze model and Partial Coherence implementation. •New Physics: Voltage-dependent carrier collection (Hecht model). •New Tool: Diode Analysis module for n-factor and Rdiff extraction. •GUI: Added Collection tab and Diode Analysis panel. 49 14.2 Version 1.5 •Added Luminescence Spectroscopy module (PL and EL) •Added standalone Shockley-Queisser Limit Calculator •Improved multijunction J-V calculation methodology •Enhanced user interface with better organisation •Performance optimisations for faster calculations •Bug fixes and stability improvements 14.3 Version 1.0.1 •Initial public release •Core TMM optical calculations •Detailed balance device modeling •SRH defect modeling framework •Batch calculations and optimisation •Layer mixing (Bruggeman EMA) 15 Appendix 15.1 List of Physical Constants Used SOLEY uses constants from the scipy.constants library (based on CODATA values): •Elementary charge (q): constants.e (≈1.602 ×10−19 C) •Planck’s constant (h): constants.h (≈6.626 ×10−34 J·s) •Speed of light in vacuum (c): constants.c (≈2.998 ×108m/s) •Boltzmann constant (kor kB): constants.k (≈1.381 ×10−23 J/K) •Electron mass (me): constants.m e (≈9.109×10−31 kg) - Used as m0for effective mass calculations. 15.2 Acknowledgments SOLEY development has benefited from extensive testing and feedback from the photovoltaics research community. The Scaffidi formulation for SRH recombination is based on work by Cuevas and colleagues. 50 15.3 Contact and Support For questions, bug reports, or feature requests: •Email: [email protected] •Include ”SOLEY” in the subject line •Provide version number (1.7) and detailed description of issue 15.4 Citing SOLEY If you use SOLEY in your research, please cite: •Software: SOLEY v1.7 (Solar Cell Optics and Electrics), Zacharie Jehl Li-Kao, 2025 •Methodology papers (to be published) 15.5 License and Terms of Use SOLEY Academic Version 1.7 is provided for research and educational purposes. Commercial use requires separate licensing. Contact the author for licensing information. 51