Semiconductor Module © Copyright 2013 COMSOL. COMSOL, COMSOL Multiphysics, Capture the Concept, COMSOL Desktop, and LiveLink are either registered trademarks or trademarks of COMSOL AB. All other trademarks are the property of their respective owners, and COMSOL AB and its subsidiaries and products are not affiliated with, endorsed by, sponsored by, or supported by those trademark owners. For a list of such trademark owners, see http://www.comsol.com/tm Outline • Introduction • The Semiconductor Module • Demonstration • Modeling Advice • Model Library • Q&A COMSOL Multiphysics 4.4 The Semiconductor Module Electrical circuit interface allows easy coupling of lumped circuits to physical device models. Specify the circuit manually or import from a SPICE netlist. Enhanced electrostatics capabilities are available within the semiconductor interface and as a standalong electrostatics interface. Heat transfer in Solids can be straightforwardly coupled to the semiconductor interface to model non-isothermal devices. Semiconductor interface implements the semiconductor equations solving Poisson’s equation and transport equations for the electrons and/or holes. The Semiconductor Interface The semiconductor interface solves the Semiconductor equations:    V   q p  n  N D  N A  n 1     J n  Un t q p 1    J p U p t q Poisson’s Equation Electron and Hole Continuity Equations Numerical methods available: • Finite Volume Method (Scharfetter-Gummel disc.) – Best current conservation • Finite Element Method (Log Formulation) – Solves for the log of the carrier concentration – Fewer degrees of freedom for the same mesh • Finite Element Method (Linear Formulation) – Provided for backwards compatibility (no longer recommended) The Semiconductor Interface Currents are defined by the drift diffusion equations: 1 J n  n n Ec   n k BTn  n n k B T 2 1 J p  p p Ev   p k BTp  p p k B T 2 Maxwell Boltzmann Carrier Statistics T J n  n n Ec   n k BTG n N c n  qnDn ,th T T J p  p p Ev   p k BTG  p N v p  qpDn ,th T Fermi Dirac Carrier Statistics Solve for : • Electrons and holes • Majority Carries only • Electrons • Holes Material Library • • Many new materials added in 4.4 Wide range of material properties available New in 4.4 • Support for arbitrary compositions Specify x to define composition Incomplete Ionization / Band Gap Narrowing • Both complete and incomplete ionization supported • Standard Model • Arbitrary user defined ionization ratios. • Arbitrary user defined models for Band Gap Narrowing MOSFET Demonstration Doping Semiconductor Doping Model Feature: • User Defined – Define a constant or functional doping form – Import data and use interpolating functions • Gaussian Doping MOSFET Demonstration (doping setup) , , Semiconductor Interface boundary conditions • Metal Contact • Ideal Ohmic (Voltage/Current/Circuit Terminal) • Ideal Schottky (Voltage/Current/Circuit Terminal) • Thin Insulator Gate (Voltage/Charge/Circuit Terminal) • Oxide layer assumed thin compared to geometry – not explicitly modeled. • Continuity/Heterojunction • Explicit (domain based) modeling of insulating regions. Can be used to model • Gates • Guard rings etc. • An array of electrostatic boundary conditions including, electric potential, ground, terminal, surface charge accumulation … Generation and Recombination Summary of the implemented recombination processes for direct (e.g. GaAs) and indirect (e.g. Si) band-gaps. Recombination: • User Defined • Direct • Shockley-Reed-Hall • Auger Generation: • Impact Ionization MOSFET Demonstration (boundary conditions) Mobility Models 1. Mobility models are added as sub-features of the Semiconductor Material model 2. A small selection of mobility models are available as pre-defined features: • Power-law (Lattice Scattering) • Arora (Impurity+Lattice Scattering) • Fletcher (Carrier-Carrier Scattering) • Lombardi (Surface Mobility) • Caughey Thomas (High Field) 3. Mobility models can be stand alone or can start from the output of another mobility model (e.g. Fletcher) 4. User defined mobility models can be mixed freely with in-built models using the same input functionality 5. The final mobility model used is selected in the semiconductor material model node. 6. Continuation settings allow non-linear mobility models to be gradually introduced to the equation system. MOSFET Demonstration (results) Supported Studies • Stationary Studies: For steady state/DC simulations • Transient Studies: For time dependent problems, e.g. turn on transients, impact ionization, etc. • Small Signal Analysis, Frequency Domain: For biased AC devices where a mix of DC and AC signals are present. Current VAC VDC Voltage Supported Studies Specific tools available for post processing small signals. Modeling Advice: Discretization Finite Volume Finite Element Log Form Constant elements - Cannot be differentiated - Special variables have been provided which cover most cases Linear or Quadratic Elements - Fully compatible with COMSOL’s framework – can use any variable anywhere in COMSOL. Current conservation implicit in the method. Energy conserving method, current conservation should be checked as part of model validation (usually OK). Stabilization not required Stabilization on by default, but not always required. Some models may solve better if this option is disabled. Finite volume variables that can be differentiated and used for coupling with other interfaces: • Variables ending with _post: e.g. semi.n_post, semi.p_post, semi.V_post • Variables associated with the electric field (E, D, the corresponding norms) • All current variables Modeling Advice: Meshing • Resolve the Debye length: , • • • The finite volume method uses flux terms at the boundary that depend on the distance to the mesh circumcenter. Elements should be minimally ‘distorted’ for accurate flux computation. Triangular and structured meshes typically perform best. In 3D using a swept mesh with a non-uniform distribution can help resolve gates better. Always assess solutions for grid independence. Modeling Advice: Solving • • • • • The semiconductor equations are highly nonlinear – this means the solvers may need some help to find the solution. Ramp up the currents and voltages in the model from zero. Use the continuation solver – set the Reuse solution for previous step setting to Auto when performing multiple sweeps. Plot the results whilst solving to get an idea of what the issue is. Use good initial values – the default will work reasonably well for equilibrium situations (no currents or thermal gradients), or use the solution from another study Modeling Advice: Solving • Use the continuity settings to gradually introduce equation terms into the system for particularly non-linear problems. • Try ramping on the doping, starting from an initially small value. • If the continuation solver does not work well – try the time dependent solver with time dependent parameters. For more detailed advice: Read the Modeling Guidelines chapter of the Semiconductor Module users guide Model Library Model Library Examples: • Bipolar Transistor • Caughey Thomas Mobility • Lombardi Surface Mobility • MESFET • MOSFET • MOSFET breakdown • MOSFET small signal • MOSFET mobility • pn diode circuit • Schottky contact • Heterojunction (1D) • pn junction (1D) Model Library: Heterojunction • • • 1D benchmark to validate the Heterojunction boundary condition. The model compares results obtained with the continuous quasi-Fermi levels and the thermionic emission model. This figure shows the benchmark result for a nGaAs/p-Al0.25Ga0.75As junction under forward and reverse bias using the thermionic emission model. Model Library: MOSFET Sequence • • Sequence of models performing different analyses on a 2D MOSFET Shows many of the important features of the semiconductor module Conclusions • Semiconductor module is now appropriate for a range of simple device simulations • Functionality has been significantly extended in version 4.4. Looking ahead • Module is currently being intensively developed • In the next year the focus will be on multiphysics applications Q&A Session