2.6.11. poisson{} (optional)

Calling this group is required if Poisson equation is to be solved during a simulation. It gathers keywords which allow choosing: initial electrostatic potential, solver with its numerical parameters, and quantities to output for obtained solution.

Initial electrostatic potential in poisson{}

It is required to specify one of 4 groups: charge_neutral{}, between_fermi_level{}, import_potential{}, or electric_field{} within the poisson{} group in the input file to choose a method for defining the initial electrostatic potential \(\phi_{i=0}\) used by the Newton solver for the Poisson equation.

poisson{ charge_neutral{} }

The recommended keyword for specifying \(\phi_{i=0}\) is charge_neutral{}. By using it, \(\phi_{i=0}\) is evaluated by requirement of charge neutrality at every point of the simulation grid. The potential is determined by solving charge neutrality equation with the bisection algorithm.

Using this group is equivalent to setting ``charge_neutral = yes`` in older syntax (until 29-06-2022).

poisson{ between_fermi_levels{} }

When this group is used then the average value of quasi-Fermi levels is taken as the \(\phi_{i=0}\) at every non-Dirichlet point of the simulation grid. Non-Dirichlet points are the grid points in the regions of the simulation, for which Dirichlet boundary conditions (in this case for potential) are not imposed. The group between_fermi_levels{} is used by default if the poisson{} group is not specified in the input file at all.

Using this group is equivalent setting ``charge_neutral = no`` in older syntax (until 29-06-2022).

poisson{ import_potential{} }

Import electrostatic potential from file or analytic function and use it as initial guess for solving the Poisson equation. If no Poisson equation is solved, the imported data determines the electrostatic potential that is used throughout the simulation, i.e. in this case an electrostatic potential can be read in that is fixed during the rest of the simulation and is used as input to the Schrödinger equation and for the calculation of the densities. The solution obtained from a problem solved previously using a different meshing is accepted.

import_from
value:

path string

Reference to imported data in import{}. The data may have more than one component (e.g. vector field).

component_number
value:

integer

default:

1

If imported data is a vector field, one may want to specify the component.

poisson{ electric_field{} }

If electric_field{} is defined, this value in units of [V] is being added to the electrostatic potential.

strength
value:

any float

default:

0.0 V/m

Defines a constant electric field in the structure. If electric_field is defined, and the absolute value is larger than zero, then it is being used for the electrostatic potential calculation.

direction
value:

3D float vector

default:

[1.0, 0.0, 0.0]

Orientation of electric field vector with respect to \((x,y,z)\) simulation coordinate system. For 1D simulations, the direction can be omitted and in this case the default will be used.

reference_potential
value:

any float

default:

0.0

Note

If poisson{} group is not called at all, then electric potential \(\phi=0\) is assumed everywhere.

Solvers in poisson{}

poisson{ newton_solver{} }

The Newton solver is used for solving the nonlinear Poisson equation. It is solved with a Newton iteration using inexact line search.The Poisson equation is nonlinear because the charge carrier density \(\rho\) depends on the electrostatic potential \(\phi\), i.e. \(\rho(\phi)\). For each Newton step a system of linear equations, \(A \cdot x = b\), is solved with a linear solver, in order to obtain a gradient. This gradient is used for the inexact line search. Generally, low temperature simulations make the Poisson equation extremely nonlinear at the beginning of the iteration and thus require more line search steps than usual. Using debuglevel = 2 displays information on the line searchs steps (search_steps): In the .log file of your simulation, you can find more information on the convergence of the Newton solver Parameters for solver of nonlinear poisson equation are as follows:

iterations
value:

any integer > 1

default:

30

Number of iterations for Newton solver

search_steps
value:

any integer between 1 and 50

default:

30

residual
value:

any float

default:

1e3 cm-2 (1D)

1e1 cm-1 (2D)

1e-4 [dimensionless] (3D)

gradient_shift
value:

any float between the interval -1e6 and 1e-6

default:

1e-13

poisson{ linear_solver{} }

Parameters for linear equation solver in Newton algorithm.

iterations
value:

any integer > 1

default:

1000

number of iterations for linear equation solver

abs_accuracy
value:

any float > 0

default:

1e1 cm-2 (1D)

1e-3 cm-1 (2D)

1e-8 [dimensionless] (3D)

rel_accuracy
value:

any float between 0.0 and 0.01

default:

1e-13

dkr_value
value:

any float < 0.5

default:

0.0

(“magic parameter” to speed up calculations, affects preconditioning. Negative values are ignored but will switch to a slightly slower but more stable preconditioner.

use_cscg
value:

yes or no

default:

no

Forces the slower but occasionally more robust CSCG (Composite Step Conjugate Gradient) linear solver to be used rather than the cg (Conjugate Gradient) linear solver. May occasionally prevent a diagonalization failure.

force_diagonal_preconditioner
value:

yes or no

default:

no

Forces the use of a slower but more robust diagonal preconditioner. Should be used only for debugging purposes, enabling will make code much slower or prevent convergence. Try setting it to yes in case preconditioning fails or the linear solver diverges. If set to yes, iterations may have to be further increased.

force_iteration
value:

yes or no

default:

no

Only for debugging purposes, enabling will make code much slower or prevent convergence

poisson{ bisection{} } (optional)

Parameters for bisection search. Used for the initial solution of the Poisson equation when charge_neutral = yes is set. Bisection is performed in order to achieve local charge neutrality at each grid point:

\[\rho = p - n + sum(N_{D,ionized}) - sum(N_{A,ionized}) = 0\]

Thus, a true classical charge neutrality is computed for classical carrier and doping situations.

Additionally, bisection is also used to determine the electrostatic potential at which contacts become charge neutral, which is also needed for ohmic contacts and charge-neutral contacts. The bisection for these contacts is performed in any case, i.e. independently to the bisection used when charge_neutral = yes is set. The bisection method is a well known algorithm for finding the root of a function. The delta is the so-called convergence tolerance parameter. Specifically in nextnano++ we use this method to find the initial solution of the Poisson equation that generally converges very fast using the default parameters and no extra tuning is required.

delta (optional)
type:

real number

unit:

eV

default:

10

constrain:

delta > 0.0

Range of bisection search.

iterations (optional)
type:

integer number

default:

40

value:

iterations > 1

residual (optional)
type:

integer number

unit:

cm-3

default:

1e3

value:

residual > 1

robust (optional)

When robust=yes then a slower charge neutrality algorithm designed to be stable for large band gaps or low temperatures.

type:

choice

values:

yes or no

default:

no

Note

The bisection algorithm is also used for initializing quasi-Fermi levels in Ohmic and charge-neutral contacts. In this case, the values specified in the input file may become internally modified. - iterations is always increased to be at least 40 - residual is reduced to be at most 1e3 cm-3 - robust is always equal yes

Therefore, the contact setup ignores bisection definitions which provide lower accuracy than these default settings.

The intrinsic density in GaN at T=300 K is of the order 1e-10 cm^-3, even smaller in AlN. Extremely low carrier densities may be also expected at low temperatures. In such cases the residual needs to be adjusted to obtain reasonable initialization of the contacts.

Attention

Reducing the default value of residual may result in significantly longer initialization times, especially in 3D simulations.

poisson{ debuglevel }

The higher this integer number, the more information on the numerical solver is printed to the screen output. Increasing the respective debuglevel to 2 or more significantly increases the volume of the diagnostic output displayed in nextnanomat (or a shell window). As result of the additional I/O load, particularly 1D simulations will slow down correspondingly (especially for current{ } and poisson{ })

value:

any integer between -1 and 3

default:

1

Outputs in poisson{}

output_potential{ }

Prints out the electrostatic potential in [eV].

output_electric_field{ }

Prints out the electric field in kv/cm.

output_electric_displacement{ }

Prints out the output electric displacement

output_electric_polarization{ }

Prints out the output electric polarization

output_dielectric_tensor{ }

Prints out the output dielectric tensor in simulation coordinate system, as it is used while setting up the sparse matrix for the Poisson solver.

boxes
value:

yes or no

default:

no

For each grid point, in 1D two points are printed out to mimic abrupt discontinuities at interfaces (in 2D four points, in 3D eight points)

output_sparse_matrix{}

Prints out the output sparse matrix used in Poisson solver.

type
type:

string list

values:

values, zero_nonzero, zero_nonzero_absolute, all

default:

values

Options for the string value list are as follows

  • values- output sparse matrix as it is (also imaginary part, if sparse matrix is complex valued)

  • zero_nonzero- output matrix containing ‘0’ and ‘1’ for zero and non-zero entries of sparse matrix (same for imaginary part, if sparse matrix is complex valued

  • zero_nonzero_absolute- output matrix containing ‘0’ and ‘1’ for zero and non-zero absolute values of entries of sparse matrix

  • all- output all types listed above