## Semiconductor Module Updates

For users of the Semiconductor Module, COMSOL Multiphysics^{®} version 5.3 brings a new *Schrödinger Equation* physics interface for quantum mechanical problems as well as several new models. Review these Semiconductor Module updates in further detail below.

### New Physics Interface: *Schrödinger Equation*

The newly added *Schrödinger Equation* interface solves the single-particle Schrödinger equation for general quantum mechanical problems in 1D, 2D, and 3D, as well as for the electron and hole wave functions in quantum-confined systems under the envelope function approximation. Appropriate boundary conditions and study types are implemented for you to set up models easily and to compute relevant quantities in various situations, such as the eigenenergies of bound states, the decay rate of quasibound states, the transmission and reflection coefficients, the resonant tunneling condition, and the effective band gap of a superlattice structure. Two new examples are included in the Semiconductor Module that help illustrate the usage of these various built-in functionalities.

**Application Library path for an example modeling the Schrödinger equation:**

*Semiconductor_Module/Verification_Examples/double_barrier_1d*

### Other Performance Improvements

#### Current-Driven Metal Contacts

A new formulation achieves easier convergence for models with current-driven metal contact boundary conditions.

#### More Options for Trap Density Specification

For each trap species type, in addition to the sum of trap densities, each individual contribution is now available as an option in the corresponding drop-down menu. In addition to traps with specified species types, traps with specified neutral energy levels can now also be defined using the same general concentration profile tools available for doping and traps.

#### User-Defined Impact Ionization Model

A user-defined model is now available for the impact ionization feature.

### New Application: Superlattice Band Gap Tool

The superlattice band gap tool helps the design of periodic structures made of two alternating semiconductor materials (superlattices). The tool uses the effective mass Schrödinger equation to estimate the electron and hole ground state energy levels in a given superlattice structure. Device engineers can use the tool to quickly compute the effective band gap for a given periodic structure and iterate the design parameters until they reach a desired band gap value.

To use the app, enter the desired superlattice parameters, including the widths of the well and the barrier layer, the effective masses for electrons and holes in those layers, the band gaps in those layers, and the conduction band offset. The valence band offset is updated automatically and should be checked for positivity by the app user. The user can also control the maximum mesh element size used for the studies. Click the Compute button to compute the shift in the conduction and valance band edge and the effective band gap. The electron and hole wave functions are plotted in the Graphics window.

**Application Library path:**

*Semiconductor_Module/Applications/superlattice_band_gap_tool*

### New Tutorial Model: Double Barrier 1D

The double barrier structure is of interest because of its application in semiconductor devices such as resonant-tunneling diodes.

This verification example demonstrates the *Schrödinger Equation* interface to set up a simple 1D GaAs/AlGaAs double barrier structure to analyze the quasibound states and their time evolution, the resonant tunneling phenomenon, and the transmission as a function of energy. The model results show very good agreement with analytical results, both for the computed eigenenergies for the quasibound states and the resonant tunneling condition, as well as the computed transmission coefficients.

**Application Library path:**

*Semiconductor_Module/Verification_Examples/double_barrier_1d*

### New Multiphysics Tutorial Model: ISFET

An ion-sensitive field-effect transistor (ISFET) is constructed by replacing the gate contact of a MOSFET with an electrolyte of interest. The concentration of a specific ionic species in the electrolyte can be determined by measuring the change in the gate voltage due to the interaction between the ions and the gate dielectric.

This tutorial of an ISFET pH sensor illustrates the procedure to set up the coupling between the semiconductor model and the electrolyte model. It also shows the technique of using a simple global equation to extract operating parameters, without the need to explicitly model the actual feedback circuitry.

*Note: In addition to the Semiconductor Module, one of the following is needed for this tutorial: the Batteries & Fuel Cells Module, Chemical Reaction Engineering Module, Corrosion Module, Electrochemistry Module, Electrodeposition Module, or the Microfluidics Module.*

**Application Library path:**

*Semiconductor_Module/Devices/isfet*

### New Tutorial Model: MOSCAP 1D

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) and computes both the low- and high-frequency C-V curves.

**Application Library path:**

*Semiconductor_Module/Devices/moscap_1d*

### New Tutorial Model: Si Solar Cell 1D

This tutorial model uses a simple 1D model of a silicon solar cell to illustrate the basic steps to set up and perform a semiconductor simulation with the Semiconductor Module. A user-defined expression is used for the photo-generation rate and the result shows typical I-V and P-V curves of solar cells.

The carrier generation mechanism from the photovoltaic effect is not modeled in detail. Instead, for simplicity, an arbitrary user-defined expression is used for the generation rate. In addition, the Shockley-Read-Hall model is employed to capture the main recombination effect. Under normal operating conditions, photo-generated carriers are swept to each side of the depletion region of the p-n junction. A small forward bias voltage is applied to extract the electrical power, given by the product of the photocurrent and the applied voltage.

**Application Library path:**

*Semiconductor_Module/Devices/si_solar_cell_1d*

### Updated Tutorial Model: Bipolar Transistor Thermal

In an update to the Thermal Analysis of a Bipolar Transistor model, the effects of a nonuniform temperature throughout a semiconductor device are investigated. The *Semiconductor* interface provides the heat source used in the *Heat Transfer in Solids* interface, while the temperature distribution that is used in the *Semiconductor* interface is calculated by the *Heat Transfer in Solids* interface. Now, the simulation can run at higher powers to reach higher temperatures, which results in more visible thermal effects.

*The voltage (top) and temperature (bottom) distributions in a bipolar transistor.*

**Application Library path:**

*Semiconductor_Module/Devices/bipolar_transistor_thermal*