OBC

Options for open-boundary condition (OBC) solvers.

The OBC solvers compute the surface Green's functions of the contacts. The surface Green's functions is the solution of the non-linear equation:

\[ \mathbf{g} = [\mathbf{M}_{0} - \mathbf{M}_{-1} g \mathbf{M}_{1} ]^{-1} \]

algorithm

algorithm: Literal['sancho-rubio', 'spectral'] = 'spectral'

The OBC algorithm to use.

• "sancho-rubio": Uses the Sancho-Rubio iterative scheme to compute the surface Green's functions. This method achieves exponential convergence compared to the linear convergence of fixed-point iterations.
• "spectral": Uses a spectral NEVP solver to compute eigenpair and uses them to construct the surface Green's functions. This is generally more efficient method when combined with a contour integral NEVP solver, but requires more parameter tuning.

nevp_solver

nevp_solver: Literal['beyn', 'full'] = 'beyn'

The NEVP solver to use for the spectral OBC algorithm.

• "beyn": Uses the Beyn's contour integral method to solve the NEVP to find the eigenpairs within a specified contour in the complex plane.

• "full": Uses a full dense eigensolver to solve for all eigenvalues by linearizing the problem. This results in a doubled problem size which is also not reduced by block sectioning / periodicity.

The following NEVP problem is solved:

\[ \sum \limits_{n=-b}^{b} \lambda^{n} \hat{\mathbf{M}}_{n} \vec{v} = 0 \]

where b goes from -block_sections to +block_sections and \(\hat{\mathbf{M}}_{n}\) are potentially reduced coupling matrices.

Only used if algorithm is set to "spectral".

block_sections

block_sections: PositiveInt = 1

The periodicity of the blocks along the transport direction.

Used in the spectral method with beyn to reduce the size of the NEVP. For example, if the supercell is constructed from 2 unit cells along the transport direction, setting this parameter to 2 will halve the size of the NEVP.

Contact blocks need to be sorted accordingly.

min_decay

min_decay: PositiveFloat = 0.001

The minimum decay rate where to differentiate between propagating and evanescent modes.

max_decay

max_decay: PositiveFloat | None = None

The maximum decay rate for evanescent modes.

Very large modes do not contribute to the surface Green's functions and can be neglected. Very large modes can also lead to numerical instabilities.

If not set, it is computed as 1.5 * log(r_o).

num_ref_iterations

num_ref_iterations: PositiveInt = 2

The number of fixed-point iterations used to refine the surface Green's functions.

\[ \mathbf{g}_{n+1} = [\mathbf{M}_{0} - \mathbf{M}_{-1} \mathbf{g}_{n} \mathbf{M}_{1} ]^{-1} \]

This is needed to improve the accuracy of the surface Green's functions if not enough eigenpairs are considered.

Only used if algorithm is set to "spectral".

min_propagation

min_propagation: PositiveFloat = 0.01

The minimum propagation speed for propagating modes.

The propagation speed is computed as:

\[ abs(real(\frac{dE}{dk})) / abs(imag(\frac{dE}{dk})) \]

residual_tolerance

residual_tolerance: PositiveFloat = 0.001

The tolerance for the residual of the eigenpairs.

The residuals are computed as:

\[ \lvert \sum \limits_{n=-b}^{b} \lambda^{b} \mathbf{M}_{n} \vec{v} \rvert \]

Modes above this tolerance are considered wrong and are not used.

Only used if algorithm is set to "spectral".

residual_normalization

residual_normalization: bool = True

Whether to normalize the residuals by the norm of the eigenvalue.

This is useful to avoid that large eigenvalues have large residuals and small eigenvalues have small residuals.

warning_threshold

warning_threshold: PositiveFloat = 0.1

The threshold for issuing a warning if the surface Green's functions residual is above this value.

The residual is computed as:

\[ \lvert \mathbf{g} - [\mathbf{M}_{0} - \mathbf{M}_{-1} \mathbf{g} \mathbf{M}_{1} ]^{-1} \rvert / \lvert \mathbf{g} \rvert \]

This parameter is only used if the formalism is wf. Otherwise, the memoizer is responsible for residual checking and warnings.

eta_decay

eta_decay: PositiveFloat = 1e-12

Small value to separate very slow decaying modes from non-decaying ones in the spectral OBC solver.

Modes that are very close to the unit contour could be misclassified with 'min_decay' and 'min_propagation' conditions i.e. when their decay is smaller than 'min_decay' but they are not propagating fast enough. The not fast enough propagating ones with decay smaller than 'eta_decay' are considered as well decaying modes.

max_iterations

max_iterations: PositiveInt = 100

The maximum number of iterations for the Sancho-Rubio method.

convergence_tol

convergence_tol: PositiveFloat = 1e-06

The convergence tolerance for the Sancho-Rubio method.

r_o

r_o: PositiveFloat = 10.0

The outer radius of the contour in the complex plane for the contour methods.

This parameter should not be too large to avoid having too many eigenpairs inside the contour. It should also not be too small to avoid missing important eigenpairs. If a eigenpair is too close to the contour, it can lead to numerical instabilities.

r_i

r_i: PositiveFloat = 0.8

The inner radius of the contour in the complex plane for the contour methods.

This parameter should be chosen to be <1 to capture propagating modes, but not too small to avoid including too many modes.

m_0

m_0: PositiveInt = 10

The subspace guess in the contour methods.

The guess has to be larger than the expected number of eigenvalues inside the contour. If too small, the method will fail. If too large, the method will be not/less efficient.

num_quad_points

num_quad_points: PositiveInt = 20

The number of quadrature points for the contour integrals.

memoizer

memoizer: MemoizerConfig

Options for memoizing the surface Green's functions.