## Semiconductor Module Updates

For users of the Semiconductor Module, COMSOL Multiphysics^{®} version 5.3a brings a new study for a quasi-Fermi level discretization option, power-driven terminals, as well as new and updated tutorials. Learn about these semiconductor features and more below.

### Semiconductor Equilibrium Study

A new study step called *Semiconductor Equilibrium* is introduced for the *Semiconductor* physics interface. You can use this new study for systems known to be in equilibrium as well as for generating initial conditions for nonequilibrium systems.

*The Model Wizard window showing the new*

The Model Wizard window showing the new *Semiconductor Equilibrium* study step.

*Semiconductor Equilibrium*study step.

### Quasi-Fermi Level Formulation

A new discretization scheme has been added using the quasi-Fermi levels as the dependent variables for the charge carriers. The quasi-Fermi level formulation provides an alternative option for tackling the often highly nonlinear equation system when modeling semiconductor devices, for example, at very low temperatures.

### Power-Driven Terminal

A new option is added to the *Metal Contact* boundary condition to specify the terminal power. This is in addition to voltage- and current-driven terminals and connecting to circuits in two different ways.

### Trapping Functionality

The functionality of the *Trapping* feature is expanded so that users can enter the initial trap occupancy and the degeneracy factor individually for each discrete or continuous energy level subnode. The energy discretization, the energy range, and number of mesh points along the energy axis can also be tailored individually for each continuous energy level subnode. The expanded functionality allows more flexibility in studying systems with complex trap properties, in particular its dynamics.

### PML for the Schrödinger Equation Interface

In addition to the *Open Boundary* condition for outgoing waves, the *Perfectly Matched Layer (PML)* functionality is added to the *Schrödinger Equation* interface to absorb outgoing waves for stationary studies. This helps the study of various scattering phenomena.

### New Tutorial Model: Gross-Pitaevskii Equation for Bose-Einstein Condensation

This tutorial model solves the Gross-Pitaevskii equation for the ground state of a Bose-Einstein condensate in a harmonic trap, using the *Schrödinger Equation* physics interface in the Semiconductor Module. The equation is essentially a nonlinear single-particle Schrödinger equation, with a potential energy contribution proportional to the local particle density. The eigenvalue study is not suitable for solving this kind of nonlinear eigenvalue problem. Instead, a stationary study is used with a global equation enforcing the normalization of the wave function to solve for the ground-state solution. The result for a large number of particles compares well with the Thomas-Fermi approximation as expected.

*Visualizing the evaporation of a Bose-Einstein condensate by solving the Gross-Pitaevskii equation for a sequence of decreasing number of particles and showing the result in an animation.*

**Application Library path:**

*Semiconductor_Module/Quantum_Systems/gross_pitaevskii_equation_for_bose_einstein_condensation*

### New Tutorial Model: MOSCAP 1D Small Signal

The metal-silicon-oxide (MOS) structure is the fundamental building block for many silicon planar devices. Its capacitance measurements provide a wealth of insight into the working principles of such devices. This tutorial constructs a simple 1D model of a MOS capacitor (MOSCAP). Both the low- and high-frequency C-V curves are computed using the approach of small-signal analysis. The model employs the quasi-Fermi level formulation and the *Semiconductor Equilibrium* study step, both new in COMSOL Multiphysics^{®} version 5.3a.

**Application Library path:**

*Semiconductor_Module/Device_Building_Blocks/moscap_1d_small_signal*

### Enhancements and Bug Fixes

- Improved finite volume formulation for incomplete ionization, spatially varying electron affinity and band gap, and consistency with the thermal equilibrium condition
- Automatic setup of the constraint value for the electric potential at a
*Metal Contact*with user-defined Schottky barrier height - Improved high field mobility model behavior at low currents
- Improved consistency of continuation parameter scaling among all doping profile types
- Fixed the formulation for band-gap narrowing (FVM and FEM), position-dependent band gap (FVM), and carrier diffusion due to temperature gradient (FVM)
- The MOSFET tutorials, which use the band-gap narrowing feature, are affected

- Corrected Fermi-Dirac variable definitions for finite element log formulation
- Fixed formulation for the case of user-defined Schottky barrier height in the
*Metal Contact*boundary condition

### Tutorial Model Improvements

#### MOSFET Series of Tutorials

The MOSFET series of tutorials are updated with a user-defined mesh that is coarser, for faster computations. This series of models includes the effect of band gap narrowing that has been improved. With the enhanced formulation, better accuracy is achieved with a coarser mesh.

#### General Model Speedup

Many tutorial models have been updated with more efficient mesh and parametric sweep settings, leading to the speedup of computation up to a factor of 10.

- MOSFET series of tutorials
- DC Characteristics of a MOS Transistor
- Breakdown in a MOSFET
- MOSFET with Mobility Models
- Small Signal Analysis of a MOSFET

- Simulation of an Ion-Sensitive Field-Effect Transistor
- GaAs p-n Junction Infrared LED
- PN-Diode Circuit
- Caughey Thomas Mobility
- Lombardi Surface Mobility
- Programming of a Floating Gate EEPROM Device
- DC Characteristics of a MESFET

#### Application Library Reorganization

The Devices category in the Application Library is replaced by four new categories:

- Device Building Blocks
- Photonic Devices and Sensors
- Quantum Systems
- Transistors