Schrödinger-Poisson

) study and study step to automatically generate the iterations in the solver sequence for the self-consistent solution of the fully coupled Schrödinger–Poisson equation. Also see the Semiconductor Module User’s Guide for the Schrödinger–Poisson Equation multiphysics interface.

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

From the list, choose (the default) or .

The ARPACK algorithm is based on an algorithmic variant of an Arnoldi process When the matrix A is symmetric it reduces to a variant of the Lanczos process. For its settings, see Study Settings for ARPACK below.

The FEAST algorithm uses an inverse residual iteration algorithm and seeks to accelerate the convergence of the subspace eigenvalue problem. For its settings, see Study Settings for FEAST. For more information about these eigenvalue solvers, see The Eigenvalue Solver Algorithms.

From the list, select a search method:

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to find all eigenfrequencies for a filled matrix. This option is only applicable for small eigenfrequency problems. You can then specify a (default: 2000).

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. 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.

By default, the physics interfaces suggest a suitable number of eigenvalues to search for. To specify the number of eigenvalues, select the check box in front of the field to specify the number of eigenvalues you want the solver to return (default: 6).

	In a 3D model, the first six eigenvalues are typically zero and correspond to the rigid-body modes of a 3D geometry. You may therefore need to specify a larger number of eigenvalues. The largest number of eigenvalues that the solver can compute is equal to the number of unconstrained degrees of freedom minus two.

By default, the physics interfaces suggest a suitable value around which to search for eigenvalues. To specify the value to search for eigenvalues around (shift), select the check box in front of the field; you can then specify a value or expression around which the eigenvalue solver should look for solutions to the eigenvalue equation (default: 0).

Use the list to control how the eigenvalue solver searches for eigenvalues around the specified shift value:

	When using a search for eigenvalues, the nonsymmetric eigenvalue solver is always used, even when the problem to solve is symmetric.

Use the field to specify the approximate number of eigenvalues you want the solver to return (default: 20). The value of the will affect the used by the algorithm; see the section of the Eigenvalue Solver. It means that increasing the value of the will increase the memory requirement and the computational time. If the solver indicates that the value of the is smaller than the actual number of eigenvalues in the given region, it will perform a search for more eigenvalues, which increases the computational time; see The Eigenvalue Region Algorithm. Within limits it is often more efficient to provide a too large value of than a too small.

In the field, you can specify a maximum number of eigenvalues to limit the eigenvalue solver’s search for additional eigenvalues (default: 200).

The check box is selected by default to increase confidence that the solver finds all eigenvalues in the search region. The work required for performing the consistency check constitutes a significant part of the total work of the eigenvalue computation.

Under , you define a unit (defaults: rad/s) and the size of the search region for eigenvalues as a rectangle in the complex plane by specifying the , , , and in the respective text fields. The search region also works as an interval method if the and are equal; the eigenvalue solver then only considers the real axis and vice versa. 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.

	The eigenvalue solver can in some cases return more than the desired number of eigenvalues (up to twice the desired number). These are eigenvalues that the eigenvalue solver finds without additional computational effort.

Eigenvalue computations can be performed with a nonsymmetric solver or, if applicable, a real symmetric solver. From the list, choose (the default) or . For the option there is the option to select the check box. This check increases the computational time and memory requirements but provides a rigorous check of the applicability of the real symmetric solver.

From the list, select a search contour:

You define the by specifying the (default: deg/s), , , and in the respective text fields. It is important that the horizontal radius that you specify in the field is large enough to enclose the eigenvalues of interest. In the field, specify the ratio of the vertical radius of the ellipse over its horizontal radius, assuming that the horizontal radius is 100. In the field, specify the rotation angle in degree from the vertical axis in the range of −180 degrees to 180 degrees.

From the list, select the method for evaluating the number of eigenvalues inside the eigenvalue search contour:

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(the default) to use stochastic estimation to evaluate the number of eigenvalues. After the stochastic estimation finishes, the eigenvalue solver automatically calculates the eigenvalues inside the eigenvalue search contour, using the number of eigenvalues calculated from stochastic estimation as the (default: 6).

In the field (default: 6), specify the initial guess of the search subspace dimension, which can be interpreted as an initial guess for the number of eigenvalues inside the contour. This setting is only available when using a stochastic estimation.

From the list, select the type for integration:

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(the default) to choose the integration type automatically depending on eigenvalue solver. It means the Gauss type for real symmetric or Hermitian eigenvalue solver and the Trapezoidal type for other types of solvers.

From the list, select the number of points for integration:

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(the default) to define the number of points for integration automatically. It is 3 for real symmetric or Hermitian eigenvalue solvers and 6 for other types of solvers.

There is also and option to select the check box. If selected, linear system factorizations are stored from the first FEAST iteration and reused in later iterations.

If the check box is selected in the dialog box, select the check box to run the FEAST eigenvalue solver in parallel. See Running FEAST in a Parallel MPI Mode for more information.

You define the by specifying the , , , and in the respective text fields.

From the list, select the method for evaluating the number of eigenvalues inside the eigenvalue search contour:

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From the list, select the type for integration:

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From the list, select the number of points for integration:

In the field (default: 6), specify the initial guess of the search subspace dimension, which can be interpreted as an initial guess for the number of eigenvalues inside the half contour. This setting is only available when using a stochastic estimation.

If required, there is an option to select the check box. There is also an option to select the check box. If selected, linear system factorizations are stored from the first FEAST iteration and reused in later iterations.

If the check box is selected in the dialog box, select the check box to run the FEAST eigenvalue solver in parallel. See Running FEAST in a Parallel MPI Mode for more information.

Use the settings under to specify a linearization point. You can specify the linearization point using the list. Select:

Use the list to specify which solution to use if has been set to . Select:

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Any other available solution to use it as a linearization point. It can be the current solution in the sequence, or a solution from another sequence, or a solution that was stored with the Solution Store node. You select a stored solution by changing to the name of the stored solution. For an eigenvalue solution, choose the eigenvalue or eigenvalues to use from the list. The default value, , use the first eigenvalue and its associated eigensolution.

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 Self-Consistent Schrödinger-Poisson Results for a GaAs Nanowire tutorial model (schrodinger_poisson_nanowire), 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 list 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 in the Semiconductor Module User’s Guide. Note that the prefix for the variable, in this case schrp1, should match the input field of the Schrödinger–Poisson Coupling multiphysics node. 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. See section Values of Dependent Variables for details.

Use the option to solve for a set of parameters. For example, in the Self-Consistent Schrödinger-Poisson Results for a GaAs Nanowire tutorial model (schrodinger_poisson_nanowire), it is used to solve for a set of azimuthal quantum numbers. See Auxiliary Sweep for details.