## Simulating the Tunneling Current Across a Graded Heterojunction

##### Chien Liu October 29, 2018

The effect of quantum tunneling can be important if the thickness of the energy barrier for the charge carrier is comparable to or smaller than the evanescent decay length. In order to account for this effect, we can use the WKB Tunneling Model feature, available in the Semiconductor Module as of version 5.4 of the COMSOL® software, for the heterojunction and Schottky contact boundary conditions. Here, we demonstrate their usage using a benchmark model.

Read More##### Chien Liu October 18, 2018

The Schrödinger-Poisson Equation multiphysics interface simulates systems with quantum-confined charge carriers, such as quantum wells, wires, and dots. Here, we examine a benchmark model of a GaAs nanowire to demonstrate how to use this feature in the Semiconductor Module, an add-on product to the COMSOL Multiphysics® software.

Read More##### Chien Liu May 31, 2017

You can use the new Schrödinger Equation interface for modeling with the Semiconductor Module in the latest release of the COMSOL® software. Let’s look at a simple example app that uses this interface to estimate the electron and hole ground state energy levels for a superlattice structure. By building apps like this one, device engineers are able to calculate the band gap for a given periodic structure and adjust the design parameters until a desired band gap value is achieved.

Read More##### Chien Liu October 20, 2015

The shortest route between two points isn’t necessarily a straight line. If by shortest route, we mean the route that takes the least amount of time to travel from point A to point B, and the two points are at different elevations, then due to gravity, the shortest route is the brachistochrone curve. In this blog post, we demonstrate how to use built-in mathematical expressions and the Optimization Module in COMSOL Multiphysics to solve for the brachistochrone curve.

Read More##### Chien Liu August 24, 2015

Today we continue our discussion on the weak formulation by looking at how to implement a point source with the weak form. A point source is a useful tool for idealizing the situation where a source is concentrated in a very small region of the modeling domain. We will find that it is very convenient to set up such a point source using the weak form.

Read More##### Chien Liu April 16, 2015

Previously in our weak form series, we discretized the weak form equation to obtain a matrix equation to solve for the unknown coefficients in our simple example problem. Following the same procedure as in this previous blog post, we will implement the equation in the COMSOL Multiphysics® software with additional steps included to examine the matrices. We will find it more convenient to use a COMSOL® software application to display all relevant matrices at once, arranged logically on one screen.

Read More##### Chien Liu April 1, 2015

Over half a century ago, Mark Kac gave an interesting lecture on a question that he had heard from Professor Bochner ten years earlier: “Can one hear the shape of a drum?” He focused on the (then undetermined) uniqueness of the set of eigenvalues given the shape of a vibrating membrane. The eigenvalue problem has since been solved and here we explore the “hearing” part of the question by considering some interesting physical effects.

Read More##### Chien Liu February 9, 2015

This post continues our blog series on the weak formulation. In the previous post, we implemented and solved an exemplary weak form equation in the COMSOL Multiphysics software. The result was validated with simple physical arguments. Today, we will start to take a behind-the-scenes look at how the equations are discretized and solved numerically.

Read More##### Chien Liu January 6, 2015

This blog post is part of a series aimed at introducing the weak form with minimal prerequisites. In the first blog post, we learned about the basic concepts of the weak formulation. All equations were left in the analytical form. Today, we will implement and solve the equations numerically using the COMSOL Multiphysics simulation software. You are encouraged to follow the steps with a working copy of the COMSOL software.

Read More##### Chien Liu November 19, 2014

This is an introduction to the weak form for those of us who didn’t grow up using finite element analysis and vector calculus in our daily lives, but are nevertheless interested in learning about the weak form, with the help of some physical intuition and basic calculus.

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