Modeling Instructions
The following steps show how to build this model using the Schrodinger Equation interface.
Model Wizard
1
Open COMSOL Multiphysics.
2
On the New page click Model Wizard () then click the 2D Axisymmetric button () on the Select Space Dimension page.
3
On the Select Physics page under Mathematics > Quantum Mechanics click Schrodinger Equation (scheq) ().
4
Click Add and then the Study button ().
5
On the Select Study page, under General Studies click Eigenvalue (). Click Done ().
Geometry Modeling
1
Under Component 1 click Geometry 1 ().
2
Add a Circle (). In the Settings window under Size and Shape, in the Radius field enter 3e-9 (3 nm). In the Sector angle field enter 180.
3
Under Position from the Base list select Center. In the r field enter 0 and in the z field enter 0, which centers the circle at the origin (these are the default settings).
4
Under Rotation Angle, in the Rotation field, enter -90 degrees to create a semicircle in the right half-plane.
5
Click Geometry 1, and add a Circle (). In the Settings window under Size and Shape, in the Radius field enter 0.5e-9 (0.5 nm). In the Sector angle field enter 180.
6
Under Position from the Base list select Center. In the r field enter 0 and in the z field enter 0, which centers the circle at the origin (these are the default settings).
7
Under Rotation Angle, in the Rotation field, enter -90 degrees to create a semicircle in the right half-plane.
8
Build the geometry by pressing F8. The final geometry consists of two semicircles in the r > 0 half-plane.
Physics Settings
1
Click the Schrodinger Equation Model node. In the Settings window the default in the Reduced mass field is the electron mass, me.
2
In the Potential energy field enter the following expression:
-e_const^2/(4*pi*epsilon0_const*sqrt(r^2+z^2))
where e_const and epsilon0_const are built-in physical constants for the electron charge and the permittivity of vacuum, respectively. sqrt(r^2+z^2) is the distance r from the origin. This expression is the potential energy in Equation 4-3.
3
Verify that the default boundary conditions are correct. Click the Axial Symmetry node and confirm it applies to the symmetry boundaries at r = 0. Click the Zero Probability node to confirm it applies to the exterior boundaries of the geometry.
Mesh Generation
The reason for the inner circular domain is to use a finer mesh in that part because the solution shows greater variations in the center region than in the outer regions for low-energy eigenvalues.
1
Right-click the Mesh node () and select Size () to add a second Size node to define the mesh size in the inner circular domain (Domain 2).
2
In the Settings window for Size 1, select Domain from the Geometric entity level list.
3
Add domain 2 to the Selection list.
4
In the Element Size section click the Custom button.
5
In the Element Size Parameters section, select the Maximum element size checkbox and enter 0.05e-9 in the corresponding field to use a mesh size no larger than 0.05 nm in domain 2.
6
Right-click the Mesh node () and select Free Triangular () to add a node that meshes domain 2 using the specified mesh size.
7
Click the top Size node. In its Settings window click to expand the Element Size Parameters section.
8
In the Maximum element growth rate field replace the default with 1.1 to make the mesh size grow more slowly toward the perimeter of the geometry.
9
In the Settings window click Build All () to create the mesh.
Computing the Solution
Specify that the eigenvalue solver should solve for 10 eigenvalues, searching around a small negative number where the first energy eigenvalues are located.
1
Under Study 1 click Step 1: Eigenvalue ().
2
In the Settings window locate the top Study Settings section.
-
In the Desired number of eigenvalues field enter 10 (after first selecting the checkbox).
-
In the Search for eigenvalues around shift field enter -2e-18 (after first selecting the checkbox).
3
In the Settings window click Compute () to start the eigenvalue solver.
Results and Visualization
By default, COMSOL Multiphysics shows a surface plot of the probability density function ⏐Ψ⏐2 for the first eigenmode. In the Settings window for the Probability Density plot, you can also click the Replace Expression button () and select Schrodinger Equation > Wave function (psi) to plot the variable psi, which is the complex-valued wave function for the electron position.
In the Settings window for the 2D Plot Group 1, you can select which eigenmode to plot. Under Data, select the corresponding eigenvalue from the Eigenvalue list. You can also verify that the eigenvalues correspond to the first energy eigenvalues listed in Results.