# currents{}¶

Specifications for the current equation such as mobility models and recombination models.

currents{ mobility_model = constant electron_mobility{ high_field_model{ vsat = 1e7 eastman{ beta = 3.0 alpha = 0.03 } } low_field_model = constant } hole_mobility{ high_field_model{ vsat = 5e6 canali{ alpha = 0.0 beta = 1.2 } } low_field_model = constant } recombination_model{ # required SRH = yes # optional, default: no Auger = yes # optional, default: no radiative = yes # optional, default: no enable_generation = yes # optional, default: yes } output_fermi_levels{} output_fermi_level_differences{} output_mobilities{} output_recombination{} output_injection{} output_currents{} output_velocities{} output_power_density{} # linear equation solver in current equation linear_solver{ iterations = 200 abs_accuracy = 1e-30 rel_accuracy = 1e-13 dkr_value = 0.1 use_cscg = no force_diagonal_preconditioner = no force_iteration = no extended accuracy = 0 } debuglevel = 1 insulator_bandgap = 0.5 # [eV] minimum_density = 1e11 # [cm^-3] minimum_density_factor = [ 1.0 , 1.0 ] # [cm^-3] maximum_density = 1e30 # [cm^-3] maximum_density_factor = [ 1.0 , 1.0 ] # [cm^-3] minimal_recombination = yes }

## Mobility models in currents{}¶

Four low-field following mobility models are supported in **nextnano++**:

constant mobility model (database{ …{ mobility_constant{} } }),

Masetti mobility model (database{ …{ mobility_masetti{} } }),

Arora mobility model (database{ …{ mobility_arora{} } }),

MINIMOS 6 mobility model (database{ …{ mobility_minimos{} } }).

One of them can be chosen by using and atribute **mobility_model**.
If neither of groups, **electron _mobility{}** or **hole_mobility{}**, is called,
then this keyword applies.

Groups **electron _mobility{}** and **hole_mobility{}** allows to define both low-field and high-field models for carriers,
separately for electrons and holes. Calling one of these groups redefines the mobility model for respective type of carriers.

Warning

Convergence may be poor or non-existent for some choices of parameters. Please make sure that the high-field model is suitable for the semiconductor system of modelled device.

- mobility_model
(optional)

- value:
“string”

- options:

`constant`

,`masetti`

,`arora`

,`minimos`

- default:

`constant`

- electron_mobility{}
(optional)& hole_mobility{}(optional)

- low_field_model

- comment:
overrides

mobility_modelfor electrons or holes- type:
choice

- options:

`constant`

,`masetti`

,`arora`

,`minimos`

- high_field_model{}
(optional)

- vsat
saturation velocity

- type:
real number

- unit:
cm/s

- constraint:
vsat >= 1.0

- haensh{}
(optional)This group triggers Haensh model. This is a default high-field model.

- canali{}
(optional)This group triggers extended Canali model

- alpha

- type:
real number

- constraint:
alpha > 0.0

- comment:
set equal 1 for Hänsch model

- beta

- type:
real number

- constraint:
beta > 0.001

- comment:
set equal 2 for Hänsch model

- transferred{}
(optional)This group triggers trensferred-electron model

- alpha
(optional)

- type:
real number

- constraint:
alpha > 0.001

- default:
1

- beta

- type:
real number

- constrain:
beta > 1.001

- gamma
(optional)

- type:
real number

- constraint:
gamma > 0.0

- default:
0

- E0

- type:
real number

- unit:
V/cm

- constraint:
E0 > 1.0

- eastman{}
(optional)This group triggers extended Eastman-Tiwari-Shur model

- alpha

- type:
real number

- constraint:
alpha > 0.0

- beta

- type:
real number

- constraint:
beta > 1.001

### High-field mobility models¶

Three high-field mobility models are currently implemented.

**Extended Canali model** given by the formula:

\(\mu(F) = \frac{ (\alpha + 1) \mu _\text{low} }{ \alpha + \left( 1 + \left( (\alpha + 1) \frac{\mu _\text{low} F}{v_\text{sat}} \right)^\beta \right)^{1/\beta} }\)

where \(\mu _\text{low}\) is low-field mobility, \(v_ \text{sat}\) is saturation velocity, and \(\mathbf{F}\) is the driving force. Parameters \(\alpha\), \(\beta\) and \(v_ \text{sat}\) are be defined independently for holes and electrons . The driving force \(\mathbf{F}\) of the respective carriers is evaluated as the gradient of the respective quasi-Fermi level.

**Hänsch model** which is a special case of the Canali model with \(\alpha=1\) and \(\beta=2\) given by the formula:

\(\mu(F) = \frac{ 2 \mu _\text{low} }{ 1 + \left( 1 + \left( 2 \frac{\mu _\text{low} F}{v_\text{sat}} \right)^2 \right)^{1/2} }\)

**Transferred-Electron model** given by the formula:

\(\mu(F) = \frac{ \mu _\text{low} + \frac{v_\text{sat}}{F}\left(\frac{F}{E_0}\right)^\beta }{ 1 + \gamma \left( \frac{F}{E_0}\right)^\alpha + \left( \frac{F}{E_0}\right)^\beta }\)

where \(E_0\) is critical field.

**Eastman-Tiwari-Shur model** given by the formula:

\(\mu\left(F\right)=\frac{\mu_\mathrm{low}+\frac{v_{\mathrm{sat}}}{F}\alpha\left(\frac{\mu_\text{low}F}{v_\text{sat}}\right)^{\beta}}{1+\alpha\left(\frac{\mu_\text{low}F}{v_\text{sat}}\right)^\beta}\)

## Recombination in currents{}¶

- recombination_model{}

- SRH
(optional)

- value:

`yes`

or`no`

- default:

`no`

Shockley-Read-Hall recombination (Shockley-Read-Hall (SRH) recombination)

- Auger
(optional)

- value:

`yes`

or`no`

- default:

`no`

Auger recombination (Auger recombination)

- radiative
(optional)

- value:

`yes`

or`no`

- default:

`no`

radiative recombination (direct recombination) (Radiative recombination)

- enable_generation
(optional)

- value:

`yes`

or`no`

- default:

`yes`

Generation can be switched on (default:

`yes`

) or off for the enabled processes.

For each *enabled* of the three processes, generation is enabled/disabled using `enable_generation`

.
Thus, enabling only generation without also enabling recombination is not possible (`enable_generation = yes`

has no effect then).
If radiative recombination is calculated (`radiative = yes`

), then the `photo_current`

is included in the file `IV_characteristics.dat`

.
Additionally, the internal quantum efficiency is written to the file `internal_quantum_efficiency.dat`

.

## Output in currents{}¶

### output_fermi_levels{}¶

prints out the Fermi levels for electrons and holes `[eV]`

### output_fermi_level_differences{}¶

prints out the difference of electron and holes Fermi levels, \(\Delta E_ \text{F} = E_{\text{F,n}} - E_{\text{F,p}}\) `[eV]`

By overlaying the Fermi level difference over the band gaps, you may e.g. determine where and involving which bands lasing may occur.

### output_mobilities{}¶

prints out the electron and hole mobilities `[cm^2/Vs]`

### output_recombination{}¶

prints out the recombination rates `[1e18/(cm^3s)]`

Note: If you want to output the generation rate, you have to specify this in `structure{`

structure{ generation{} & injection{} } `==>`

`output_generation{} }`

.

### output_currents{}¶

prints out the electron and hole current densities `[A/cm^2]`

Note: The electron, hole, and total currents (integrated over the contacts surfaces) are always written into the files
`IV_electrons.dat`

, `IV_holes.dat`

, and `IV_characteristics.dat`

(in units of `[A/cm^2]`

(1D), `[A/cm]`

(2D), `[A]`

(3D)), respectively.
If radiative recombination is used, the file `IV_characteristics.dat`

also contains the photo current.
In all `IV_*.dat`

files, the first columns indicate the voltages at each contact.
Typically, the *first* column should be the one that is swept as it is then easier to plot the results within nextnanomat as the first column is the x axis in such a plot.
You can switch the columns by reordering the contacts, see contacts{}.
The consumed power is written in `IV_Power.dat`

in units of `[W/cm^2]`

(1D), `[W/cm]`

(2D), `[W]`

(3D), respectively.
The *emitted power* column is added if the energy resolved density integration is enabled.

### output_velocities{}¶

prints out electron and hole drift velocities `[cm/s]`

### output_forces{}¶

prints out driving forces of electrons and holes `[eV/nm]`

### output_power_density{}¶

prints out power density (only Joule heating) `[W/cm^3]`

## Solver in currents{}¶

### linear_solver{}¶

Parameters for linear equation solver in current equation.

- iterations

- value:
integer

- default:

`10000`

- example:

`200`

number of iterations

- abs_accuracy

- value:
double

- default:

`1e-30`

#`[eV]`

- example:

`1e-30`

#`[eV]`

absolute accuracy of Fermi level, use a small value to force convergence

Note

abs_accuracy is

(dimension dependent!!!):(default is: 1e-30 [eV] (1D), 1e-30 [eV] (2D), 1e-30 [eV] (3D))

- rel_accuracy

- value:
double,

`0.0`

<=`rel_accuracy`

<=`0.01`

- default:

`1e-13`

# [dimensionless]relative accuracy

(default is: 1e-13 [dimensionless]), 0.0 <= rel_accuracy <= 0.01

- dkr_value

- value:
double, must be <= 0.5

- default:

`-1.0`

# [dimensionless]- example:

`0.1`

# [dimensionless]

magic parameterto speed up calculations, affects preconditioningNote: 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`

Only for debugging purposes, enabling will make code much slower or prevent convergence. Forces the use of a slower but more robust diagonal preconditioner. Only for debugging purposes, enabling will make code much slower or prevent convergence. Please 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.

- extended_accuracy

- value:

`0`

or`1`

- default:

`0`

If set to

`1`

, then current equation is solved using slower but more accurate solver (only implemented for nonperiodic 1D simulations). Experimental feature, will change in the future.

### debuglevel¶

- value:
integer value between [-1,3]

- default:
1

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 currents{} and poisson{}).

### insulator_bandgap¶

- value:
double

- default:

`1.0`

#`[eV]`

- example:

`0.5`

#`[eV]`

\(I_{\text{gap}}\) affects initial solution of Fermi level.

A large value (relative to band gap) of \(I_{\text{gap}}\) lets Fermi level drop continuously.
A small value of \(I_{\text{gap}}\) lets Fermi level drop in barrier and makes it flat in small bandgap regions.
A better, more meaningful, name for `insulator_bandgap`

might have been `initial_energy_scale`

.

The drift-diffusion current equation reads \(\text{div} \left( \mu n \nabla E_{\text{F}} \right) = G-R\).
In order to calculate the density \(n\), we have to know the quasi-Fermi level \(E_{\text{F}}\).
Approximately, the intrinsic density exponentially depends on the band gap \(E_{\text{gap}}\).
Therefore we apply a trick, and use \(\text{div} \exp \left( E_{\text{gap}} / I_{\text{gap}} \right) \nabla E_{\text{F}} = 0\) in order to find a first approximation to the quasi-Fermi level,
where \(I_{\text{gap}}\) can be entered in the input file (`insulator_bandgap`

) to adjust the convergence behavior of the initial solution.

### minimum_density¶

- value:
double

- default:

`1e10`

#`[cm^-3]`

- example:

`1e11`

#`[cm^-3]`

Improves condition number of current equation matrix.

Note: `1e12`

`[cm^-3]`

seems to be too high.

`1e-10`

is actually the smallest possible value, smaller values are just automatically increased to `1e-10`

without warning.
Minimum charge carrier density (lower limit) for both electrons and holes that can appear in drift-diffusion current equations.
The minimum density might have to be increased in order to obtain convergence for the drift-diffusion current equations.
The minimum density should be as low as possible.
The minimum density can be chosen as large as possible but should be smaller than the minimum density in the converged result.
As the drift-diffusion current is proportional to the charge carrier density, this eventually also sets the lower limit of the current.
The minimum density is a useful flag for structures where regions are present that have almost no density (e.g. a barrier, or insulator).
If the density in such an insulator is below `1 - 10^3`

`[cm^-3]`

,
the product of \(\mu n\) in the drift-diffusion current equation varies over several orders of magnitude.
Consequently, the matrix used in the linear solver is not well conditioned.
Here, the current through these insulating regions is basically zero which has implications on the convergence behavior of the drift-diffusion current equations.
Increasing the minimum density will help in these cases.
A useful value for the minimum density of a certain material depends on the band gap because its intrinsic density also depends on the band gap.
A wide-band gap material has a much lower intrinsic density than a low-band gap material.

Minority carriers in highly doped semiconductors or carriers in undoped wide band gap semiconductors have extremely small equilibrium densities (much less than `1.0 cm^-3`

),
resulting in complete breakdown of the solvers for current equation due to underflow.
Unfortunately, it is not clear whether quadruple precision would cure this problem.
Also, it seems unphysical to believe that one carrier per kilometre can be physically relevant (aside from the rather practical issue that real-life minority carrier densities are not in thermal equilibrium und thus never become as small as predicted).
Therefore, we modified the code (2019-01-23) to make sure that the minimum density parameter as specified for the current equation is not smaller than `10^-10 cm^-3`

(this value corresponds to a conductivity 10 orders of magnitude lower than the best insulators).
At the same time, we left the minimum value in the syntax definition unchanged as `0.0`

in order to avoid users getting annoying error messages when they experimentally set the value to zero (zero will just be silently increased to `1e-10`

).
For systems without extremely low carrier densities, results will not be affected by this change.

Separate minimum densities for electrons and holes are optionally available:

## minimum_density_factor¶

- value:
double array

- default:

`[ 1.0 , 1.0 ]`

#`[cm^-3]`

- example:

`[ 1.0 , 0.01 ]`

#`[cm^-3]`

Here, the two numbers for `minimum_density_factor`

defines scaling factors by which the minimum density is multiplied for *electrons* and for *holes*.
For example, a minimum density of `1e10`

`[cm^-3]`

for *electrons* and `1e8`

`[cm^-3]`

for *holes* could be defined e.g. as

minimum_density = 1e10 # [cm^-3] minimum_density_factor = [1.0 , 0.01] # [cm^-3]

or

minimum_density = 1e9 # [cm^-3] minimum_density_factor = [10 , 0.1] # [cm^-3]

or, most conveniently,

minimum_density = 1.0 # [cm^-3] minimum_density_factor = [1e10 , 1e8] # [cm^-3]

Irrespective of the definition, the actually used values for the minimum densities are output close to the beginning of the log file.

## maximum_density¶

- value:
double

- default:

`1e30`

#`[cm^-3]`

similar meaning as `minimum_density`

## maximum_density_factor¶

- value:
double array

- default:

`[ 1.0 , 1.0 ]`

#`[cm^-3]`

analogous to `minimum_density`

Note

Note that the minimum/maximum densities only affect the current operators (\(\nabla \mu n \nabla\)) and
corresponding currents (\(e \mu n \nabla\)), thus,
they have no direct influence on recombination (unless `minimal_recombination = yes`

), computed densities, Poisson equation, etc.
Also, when restricting effective densities in the current equations from above and/or below, please think first whether and in which way such restriction affects the physics:
Increasing minimum densities make insulating regions less insulating, whereas decreasing maximum densities make conducting regions less conducting.

## minimal_recombination¶

- value:

`yes`

or`no`

- default:

`no`

If enabled, the minimum densities will also apply to the recombination/generation terms of the current equation.