qtcad.device.schrodinger module

Schrodinger solver.

class qtcad.device.schrodinger.Solver(d, solver_params=None)

Bases: Solver

Finite element solver for Schrodinger’s equation.

Attributes:

  • d (device object) – The device for which to solve Schrodinger’s equation

  • num_states (int) – Number of states to solve for, starting from the lowest eigenvalue.

  • tol (float) – Solver tolerance (stopping criterion). Relative accuracy for eigenvalues.

  • maxiter (int) – Maximum number of iterations.

  • method (string) – Method to use for the solution of the eigenvalue problem. ‘robust’ or ‘fast’.

  • verbose (bool) – Verbosity level.

  • mass_tensor (2D 3x3 array or None) – Inverse mass tensor to be used in Schrodinger equation eigenvalue problem.

Note

When using electrons (conf_carriers = ‘e’), the Hamiltonian is constructed using either the effective mass matrix of the device, or the effective mass specified in the solver parameters.

When using holes (conf_carriers = ‘h’), the D matrix of the device is used. The D matrix must be written in units of \(\hbar^2/m_e\), and must lead to positive dispersion in k space.

tol and maxiter are None by default, meaning that they are left to the default values in the SciPy solvers.

__init__(d, solver_params=None)

Initialize from device object d.

Parameters:

  • d (Device object) – The device to simulate.

  • solver_params (SolverParams obj or None, optional) – The parameters of the solver.

band_structure(K, basis_vec=None, mixing=True)

Computes the band structure for some translationally invariant problem

Parameters:

  • K (1d array) – Magnitudes to multiply the basis vector by to obtain the wavevectors for which we wish to compute the band structure

  • basis vector (1d array) – Direction along the translationally invariant directions. Defaults to None, in which case the basis vector is taken to be one with entries of 1 along each translationally invariant direction.

  • mixing (bool) – If False, the band off-diagonal matrix elements of the new a matrix will be set to 0

Returns:

2d array – Energies from the band structure calculation. First index runs over the k values and the second over the energy levels.

solve(k=None, mixing=True)

Solve Schrodinger’s equation.

Parameters:

  • k (1d numpy array) – k vector along translationally invariant directions. This array should have as many components as the length of the attribute ti_directions. Defaults to None in which case it is taken to be an array of zeros.

  • mixing (boolean) – Whether to consider band off-diagonal elements in the problem.

Note

Modifies: self.d.energies and self.d.eigenfunctions.

class qtcad.device.schrodinger.SolverParams(inp_dict=None)

Bases: SolverParams

Parameters to pass to a Schrödinger solver.

Attributes:

  • tol (float) – Tolerance to use in a calculation, in eV. Default: 1e-6.

  • maxiter (int) – Maximum number of iterations in the fast eigenvalue solver. Default: 1000.

  • errortype (str) – Type of error to use in a convergence criterion. Relative (‘rel’ or ‘relative’) or absolute (‘abs’ or ‘absolute’). Default: ‘abs’.

  • method (str) – The method to use to the eigenvalue problems: “robust” or “fast”. Default: “fast”.

  • parallel (bool) – Whether to use parallel algorithms to solve the eigenvalue problem. Only available for the “fast” method. The parallel algorithm is typically faster than the serial algorithm when (1) the system has at least 4 available logical cores and (2) the (sub)mesh on which the Schrödinger equation is solved has at least 10,000 nodes. Default: True.

  • num_states (int) – Number of states to solve for, starting from the lowest eigenvalue. Default: 10.

  • guess (tuple, str, or None) – Initial guess to use in the first Schrödinger iteration when the fast solver is used. If None, use default guess (random). If “box”, uses the particle-in-a-box eigenfunctions. Otherwise, must be a tuple whose first entry is the wave functions (with global node index first, and state index second), and whose second entry is the eigenenergies. Default: None.

  • mass_tensor (2D 3x3 array or None) – Inverse mass tensor to be used in Schrodinger equation eigenvalue problem that supersedes the mass tensor from the device. Default: None, meaning that the mass tensor of the device is used.

  • ti_directions (list) – List of strings labeling directions (‘x’, ‘y’, or ‘z’) along which translational invariance is assumed by the Schrödinger solver. Default: None.

  • verbose (bool) – Verbosity level. Default: False.

__init__(inp_dict=None)

Instantiate the SolverParams object.

Parameters:

inp_dict (dict, optional) – Dictionary specifying certain solver parameters to be used instead of their default values.