Schrödinger–Poisson Study Step

The dedicated Schrödinger–Poisson study step automatically generates the iterations in the solver sequence for the self-consistent solution of the fully coupled Schrödinger–Poisson equation.

To take advantage of the default settings detailed below, use the or the button in the ribbon/toolbar of the graphical user interface. A study step added by right-clicking a Study node does not include any default settings suggested by the physics.

The default option for the list is , because for a completely new problem, it is often necessary to use this option to find the range of the eigenenergies and a rough estimate of the number of eigenstates. Make sure that the list is left as blank (default).

Once the range and number are found, switch to the search option with appropriate settings for the range and number of eigenvalues, in order to ensure that all significant eigenstates are found by the solver.

Make sure the list is left as blank (default).

The real and imaginary parts of the input fields refer to the real and imaginary parts of the eigenvalue, respectively (not the real and imaginary part of the eigenfrequency). So, to look for the eigenenergies of bound states, set the input fields for the real parts to the expected energy ranges, and set the input fields for the imaginary parts to a small range around zero to capture numerical noise or slightly leaky quasi-bound states.

If there are extra domain or boundary conditions used to obtain the initial value for the fully coupled problem, remember to disable them here. For example, see the schrodinger_poisson_nanowire tutorial model, where the Thomas–Fermi solution is used as the initial condition, and the space charge density contribution from the Thomas–Fermi approximation is disabled here.

The default option for the drop-down menu is , which updates a table displaying the history of a global error variable after each iteration during the solution process. This provides a good indication of the solution process to monitor whether the iteration is converging.

The default expression for the input field uses the built-in global error variable schrp1.global_err, which computes the max difference between the electric potential fields from the two most recent iterations, in the unit of V, as discussed in the section Charge Density Computation for the Schrödinger–Poisson Coupling multiphysics node earlier. Note that the prefix for the variable, in this case schrp1, should match the input field of the Schrödinger–Poisson Coupling multiphysics node. (See section Settings) Setting the to 1e-6 thus means the iteration ends after the max difference is less than 1 uV.

Use this section to configure the initial conditions for the study step. Consult the COMSOL Multiphysics Reference Manual for details.

Use the option to solve for a set of parameters, for example, in the schrodinger_poisson_nanowire tutorial model, it is used to solve for a set of azimuthal quantum numbers. The total particle density will be computed using the statistically weighted sum of all the eigenstates for all parameters.