Introduction
Device engineers and physicists use the Semiconductor Module to design and optimize semiconductor devices. For many years semiconductor device design has been closely associated with the use of simulation tools, due to the high cost of prototyping new devices and processes. The advance of nanotechnology and organic semiconductors has helped create many novel devices. Researchers in these fields also use simulation to assist the fundamental understanding and design optimization. For all of the systems mentioned above, multiphysics effects often play an important role, and COMSOL Multiphysics is the ideal platform for investigating these effects.
The Semiconductor Module includes a predefined Semiconductor interface, which is based on the conventional drift–diffusion formulation. An optional density-gradient implementation is also available to provide a computationally efficient method to add the effect of quantum confinement to the drift–diffusion equation system.
In addition, a predefined Schrödinger Equation interface and a predefined Schrödinger–Poisson Equation multiphysics interface allow more detailed modeling of quantum-confined systems such as quantum wells, wires, and dots.
The Semiconductor Module enables the stationary and dynamic performance of devices to be modeled in one, two, and three dimensions, together with circuit-based modeling of active and passive devices. In the frequency domain, it is possible to model devices driven by a combination of AC and DC signals. A broad range of semiconductor devices can be modeled, and phenomena such as heat generation, electrochemical reaction, and optoelectronic effects can be straightforwardly included using predefined or manual couplings.
The Semiconductor interface solves the drift–diffusion and Poisson’s equations by either the finite volume or the finite element method. The physics interface solves a set of coupled partial differential equations for the electric potential and for the electron and hole concentrations (or their logarithm in the case of the finite element method log formulation, or their quasi-Fermi levels in the case of the quasi-Fermi level and density-gradient formulation). The corresponding initial and boundary conditions are easily specified in the physics interface.
The COMSOL Multiphysics design emphasizes the physics by providing users with the equations solved by each feature and by offering full access to the underlying equation system. The Semiconductor Module User’s Guide provides complete information on the theory underlying the Semiconductor interface.
There is also tremendous flexibility to add user-defined equations and expressions to the system. For example, user-defined mobility models can be readily specified simply by typing appropriate expressions into the user defined feature, no scripting or coding is required. These user-defined mobility models can be combined arbitrarily with the predefined mobility models built into the software. When COMSOL Multiphysics compiles the equations, the complex couplings generated by these user-defined expressions are automatically included in the equation system. The equations are then solved using a range of state-of-the-art solvers.
Once a solution is obtained, a large range of result analysis tools are available to interrogate the data, and predefined plots are automatically generated to show the device response. COMSOL Multiphysics offers the flexibility to evaluate a wide range of physical quantities including predefined quantities such as the electron and hole currents (including current components from drift, diffusion, and thermal diffusion), the electric field, the generation–recombination rate, and the temperature, all available through easy-to-use menus, as well as arbitrary user-defined expressions.
To model a Semiconductor device, the geometry is first defined in the software. Then appropriate materials are selected and the Semiconductor interface is added. The dopant distribution can be computed separately using a diffusion equation calculation, imported from third party software, or specified empirically using the built-in doping features. Initial conditions and boundary conditions are set up within the physics interface. Next, the mesh is defined and a solver is selected. Finally the results are visualized using a wide range of plotting and evaluation tools. All of these steps are accessed from the intuitive COMSOL Desktop graphical user interface.
The Schrödinger Equation interface solves the Schrödinger equation for the wave function of a single particle in an external potential. This can be applied to general quantum mechanical problems, 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 the user to easily set up models to compute relevant quantities in various situations, such as the eigenenergies of bound states, the decay rates of quasi bound states, the transmission and reflection coefficients, the resonant tunneling condition, and the effective band gap of a superlattice structure. The Double Barrier 1D, Superlattice Band Gap Tool, and k dot p Method for Strained Wurtzite GaN Band Structure models in the Application Libraries illustrate the usage of the various built-in functionalities.