# $Monte-Carlo¶

Monte Carlo transport: Here, the parameters for the Monte Carlo procedure and the mobility calculations are defined.

```
$Monte-Carlo optional !
Monte-Carlo-transport character required !
destination-directory character required !
grid-position double_array required ! [nm]
energy-dispersion character required !
charge-carrier-type character required !
number-of-simulated-carriers integer required !
number-of-timesteps integer required !
timestep double required ! [fs]
number-of-electric-field-steps integer required !
electric-field-start-value double_array required ! [V/m]
electric-field-step double required ! [V/m]
electric-field-step-factor double required !
start-averaging-after-timesteps integer required !
execute-averaging-after-timesteps integer required !
impurity-background-doping-concentration double required ! [cm-3]
doping-concentration double required ! [cm-3]
LA-phonon-scattering character optional !
LO-phonon-scattering character optional !
TA-phonon-scattering character optional !
TO-phonon-scattering character optional !
acoustic-phonon-scattering character optional !
polar-optical-phonon-scattering character optional !
plasmon-scattering character optional !
ionized-impurity-scattering character optional !
surface-roughness-scattering character optional !
electron-hole-scattering character optional !
impact-ionization-scattering character optional !
!
alloy-scattering character optional !
alloy-disorder-scattering-potential double optional ! [eV]
!
mass-density double optional ! [kg/m3]
sound-velocity double optional ! [m/s]
acoustic-deformation-potential double optional ! [eV]
!
quantum-well-width double optional ! [nm]
spacer-width double_array optional ! [nm]
remote-doping-sheet-density double_array optional ! [cm-2]
2DEG-sheet-density double optional ! [cm-2]
2DEG-sheet-density-number-of-subbands double optional !
$end_Monte-Carlo optional !
```

*Syntax*

```
destination-directory = Monte_Carlo/
```

```
Monte-Carlo-transport = yes ! Monte Carlo transport calculations switched on
= no ! Monte Carlo transport calculations switched off
= simple ! Monte Carlo transport calculations switched off and a simple 2DEG algorithm is employed
```

```
grid-position = 10.0 ! grid position in units of [nm]
! take material parameters from the material located at this point
grid-position = 10.0 ! x = 10 [nm] (1D)
= 10.0 20.0 ! x = 10 [nm], y = 20 [nm] (2D)
= 10.0 20.0 20.0 ! x = 10 [nm], y = 10 [nm], z = 20 [nm] (3D)
```

```
energy-dispersion = kp ! use nonparabolic and anisotropic energy dispersion E(k) calculated from k.p theory, i.e.
! take into account k.p dispersion into scattering rates, overlap factors and selection of final states
= parabolic ! use parabolic and isotropic energy dispersion E(k) (single-band effective-mass approximation), i.e.
! take into account parabolic dispersion into scattering rates, overlap factors and selection of final states
```

```
charge-carrier-type = holes ! Monte Carlo transport calculations for holes
= electrons ! (not implemented yet)
```

```
number-of-simulated-carriers = 10000 ! number of simulated electrons or holes, respectively
! (depending on charge-carrier-type = ...)
```

```
number-of-timesteps = 600 ! number of timesteps
timestep = 10.0 ! timestep in units of [fs]
```

total time = `number-of-timesteps`

`*`

`timestep`

= 600 `*`

10 fs = 6000 fs = 6 ps

```
number-of-electric-field-steps = 5 ! number of calculated points for increasing electrical field (starting value: 1 kV/cm)
electric-field-start-value = 1.0e5 0.0 ! [V/m], F = (Fx,Fy), i.e. in this case Fx = 1.0 * 10^5 V/m = 1 kV/cm,
! Fy = 0 V/m
electric-field-step = 1.0e3 ! increase electric field strength along the channel by this magnitude (for each electric field step)
! [V/m], i.e. in this case 1.0 * 10^5 V/m, Units: 1 kV/cm = 1 * 10^5 V/m = 1e5 V/m
! (1.0d3 = 0.01 kV/cm = 1000 V/m = 1 kV/m = 10 V/cm)
electric-field-step-factor = 1.1 ! increase electric field strength along the channel by this factor (for each electric field step)
= 1.0 ! [] linear increase
```

\(F_{x,i+1} = F_{x,i}\) `*`

`electric-field-step-factor`

+ `electric-field-step`

\(F_{y,i+1} = F_{y,i}\) where \(F_{x}\) and \(F_{y}\) are the electric field magnitude along the \(y\) or \(y\) direction, respectively.

```
start-averaging-after-timesteps = 400 ! number of timesteps after which the averaging mechanism is started
execute-averaging-after-timesteps = 1 ! number of timesteps after which the averaging mechanism is executed
! (= 1: each timestep is used for averaging)
```

```
impurity-background-doping-concentration = 0.5e15 ! in units of [cm^-3] ==> 0.5e15 = 0.5 * 10^15 cm-3
```

The impurity background doping concentration which is relevant for the impurity scattering mechanism is given in units of [cm^-3]. Doping with fully ionized dopants is assumed in this case (during the Monte Carlo procedure only). The whole simulation region will be covered with this background doping during the Monte Carlo procedure. It will contribute additively to the additional doping regions specified inside the input file ($doping-function). It also enters the scattering rates for impurity scattering: If it is set to zero, the integral will not converge, so don’t do it…

```
doping-concentration = 1e18 ! in units of [cm^-3] ==> 1e18 = 1 * 10^18 cm^-3
```

The doping concentration is given in units of [cm^-3]. This value for the doping concentration is used for the impurity scattering table. It is relevant for electron-hole scattering and ionized impurity scattering.

*Flags to switch on/off certain scattering models*

```
LA-phonon-scattering = yes ! include LA phonon scattering (default)
= no ! switch off LA phonon scattering
```

This affects the subroutines

total scattering probability due to LA phonon emission in first conduction band

total scattering probability due to LA phonon emission among hole bands

total scattering probability due to LA phonon absorption in first conduction band

total scattering probability due to LA phonon absorption among hole bands

```
LO-phonon-scattering = yes ! include LO phonon scattering (default)
= no ! switch off LO phonon scattering
```

This affects the subroutines

total scattering probability due to LO phonon emission in first conduction band

total scattering probability due to LO phonon emission among hole bands

total scattering probability due to LO phonon emission among 2D hole subbands

total scattering probability due to LO phonon absorption in first conduction band

total scattering probability due to LO phonon absorption among hole bands

total scattering probability due to LO phonon absorption among 2D hole subbands

```
TA-phonon-scattering = yes ! include TA phonon scattering (default)
= no ! switch off TA phonon scattering
```

This affects the subroutines

total scattering probability due to TA phonon emission in first conduction band

total scattering probability due to TA phonon emission among hole bands

total scattering probability due to TA phonon absorption in first conduction band

total scattering probability due to TA phonon absorption among hole bands

```
TO-phonon-scattering = yes ! include TO phonon scattering (default)
= no ! switch off TO phonon scattering
```

This affects the subroutines

total scattering probability due to TO phonon emission in first conduction band

total scattering probability due to TO phonon emission among hole bands

total scattering probability due to TO phonon absorption in first conduction band

total scattering probability due to TO phonon absorption among hole bands

```
acoustic-phonon-scattering = yes ! include acoustic phonon scattering (default)
= no ! switch off acoustic phonon scattering
```

This affects the subroutines

total scattering probability due to acoustic scattering in first conduction band

total scattering probability due to acoustic scattering among hole bands

total scattering probability due to acoustic scattering among 2D hole subbands

The acoustic phonon scattering rates are linear functions of temperature.

```
polar-optical-phonon-scattering = yes ! include polar optical phonon scattering (default)
= no ! switch off polar optical phonon scattering
```

This affects the subroutines

total scattering probability due to polar optical phonon emission

total scattering probability due to polar optical phonon absorption

```
plasmon-scattering = yes ! include plasmon scattering (default)
= no ! switch off plasmon scattering
```

This affects the subroutines

total scattering probability due to heavy hole - plasmon absorption in first conduction band

total scattering probability due to Gamma - plasmon absorption in hole bands

total scattering probability due to heavy hole - plasmon emission in first conduction band

total scattering probability due to Gamma - plasmon emission in hole band

```
ionized-impurity-scattering = yes ! include TO phonon scattering (default)
= no ! switch off TO phonon scattering
```

This affects the subroutines

total scattering probability due to ionized impurities

total scattering probability due to ionized impurity scattering among 2D hole subbands

```
surface-roughness-scattering = yes ! include surface roughness scattering (default)
= no ! switch off surface roughness phonon scattering
```

This affects the subroutines

total scattering probability due to surface roughness scattering among 2D hole subbands

```
electron-hole-scattering = yes ! include electron-hole scattering (default)
= no ! switch off electron-hole scattering
```

This affects the subroutines

total scattering probability due to electron-hole scattering in first conduction band

total scattering probability due to electron-hole scattering in hole bands

```
impact-ionization-scattering = yes ! include impact ionization scattering (Kane model) (default)
= no ! switch off impact ionization scattering (Kane model)
```

This affects the subroutines

total scattering probability due to impact ionization scattering (Kane model)

```
alloy-scattering = yes ! include alloy scattering (default)
= no ! switch off alloy scattering
```

This affects the subroutines

total scattering probability due to alloy scattering in first conduction band

total scattering probability due to alloy scattering among hole bands

```
alloy-disorder-scattering-potential = 1.0 ! [eV] (optional parameter)
= 1.0 ! [eV] 1 eV for AlGaAs
= 0.6 ! [eV] ~0.6 eV for InGaAs
```

If `alloy-disorder-scattering-potential`

is not present, then this value will be calculated internally from the conduction (or valence) band offset of the two binary end points for each grid point, e.g. Al(x)Ga(1-x)As: CBO(AlAs) - CBO(GaAs)
Some more information about the used parameters for each grid point is contained in the file `AlloyScatteringInfo.dat`

.

*Material parameters*

The following parameters are used for deformation potential acoustic phonon scattering.

```
mass-density = 5.79e3 ! [kg/m3] lattice density
sound-velocity = 3.7e3 ! [m/s] velocity of longitudinal elastic waves
acoustic-deformation-potential = 7.2 ! [eV]
```

The following parameters are used for the *simple algorithm* (for delta-doped 2DEGs) to calculate the mobility.

```
quantum-well-width = 10.0 ! [nm]
spacer-width = 20.0 ! [nm]
remote-doping-sheet-density = 1e12 ! [cm-2]
2DEG-sheet-density = 0.071e12 ! [cm-2]
2DEG-sheet-density-number-of-subbands = 3 ! (default: 1)
```

If `2DEG-sheet-density`

is not present, then the 2DEG density is calculated automatically where the number of subbands that are taken into account can be specified (default: `1`

).
If two remote doping regions should be taken into account, one can input an array of values.

```
spacer-width = 20.0 10.0 ! [nm] spacer width of first and 2nd doping region
remote-doping-sheet-density = 1e12 1e11 ! [cm-2] remote doing density of first and 2nd doping region
```

*Monte Carlo*

The Monte Carlo algorithm that is used inside *nextnano³* is similar to the following publication:

Subband structure and mobility of two-dimensional holes in strained Si/SiGe MOSFET’sR. Oberhuber, G. Zandler and P. VoglPhysical Review B 58, 9941 (1998)

*Details*

The carrier transport in quantized 2D channels is computed in terms of a momentum space ensemble Monte Carlo procedure for a spatially homogeneous channel with constant electric fields invoking consistently calculated 2D scattering rates (see subroutine `scattering`

for details on scattering).

For electrons and a single conduction band, a similar procedure was carried out previously (M.V. Fischetti, S.E. Laux, PRB 48, 2244 (1993)).

The equations of motion are integrated with the numerically determined, fully nonparabolic subband dispersions E(k) in analogy to full band Monte Carlo methods that have been developed previously for bulk (M.V. Fischetti, S.E. Laux, PRB 38, 9721 (1988)).

Both the hole dispersion relations E(k) and the inverse relation k(E) have been determined by a 2D discretization of k space.

The Monte Carlo method requires both the individual as well as the total scattering rates. The latter can only be obtained, for each initial state in subband … with energy E(k), by a numerical integration of Eq. (…) over all final states. This amounts to integrating over the wavevectors q,q with the numerically determined 2D band structure.

*Restrictions*: So far, this Monte Carlo procedure only makes sense for 1D simulations with quantum confinement.

Note: Do not forget to specify the parameters for the density of states (DOS) calculation. The DOS is used for the calculation of the scattering tables.

```
$output-kp-data
...
DOS-density-of-states = ...
DOS-Emin-Emax = ...
DOS-points = ...
```

## Scattering mechanisms¶

polar optical phonon emission

ionized impurities

polar optical phonon absorption

LO phonon emission in first conduction band

LO phonon emission among hole bands

LO phonon emission among 2D hole subbands

LO phonon absorption in first conduction band

LO phonon absorption among hole bands

LO phonon absorption among 2D hole subbands

TO phonon emission in first conduction band

TO phonon emission among hole bands

TO phonon absorption in first conduction band

TO phonon absorption among hole bands

LA phonon emission in first conduction band

LA phonon emission among hole bands

LA phonon absorption in first conduction band

LA phonon absorption among hole bands

TA phonon emission in first conduction band

TA phonon emission among hole bands

TA phonon absorption in first conduction band

TA phonon absorption among hole bands

acoustic scattering in first conduction band

acoustic scattering among hole bands

acoustic scattering among 2D hole subbands

alloy scattering in first conduction band

alloy scattering among hole bands

electron-hole scattering in first conduction band

electron-hole scattering in hole bands

heavy hole plasmon absorption in first conduction band

Gamma plasmon absorption in hole bands

heavy hole plasmon emission in first conduction band

Gamma plasmon emission in hole band

surface roughness scattering among 2D hole subbands

ionized impurity scattering among 2D hole subbands

### Output¶

*Output files for k.p data*

*Density of states (DOS)*

The two-dimensional density of states (DOS) is written to the files `Schroedinger_kp/DOS_hl_6x6kp.dat`

and `DOS_hl_6x6kp_sum.dat`

(and similar for electrons).
The DOS has been calculated from the energy dispersion \(E({\mathbf k}) = E(k_x,k_y)\) and is used inside the Monte Carlo procedure (e.g. scattering tables).

For details, see $output-kp-data.

*Output files for transport data*

After the total simulation time is reached, the averaged transport quantities are written to the output files with the subroutine `write_outputMC`

.
This is done for each value of the applied electric field.
The results of the calculated transport quantities are stored in the files:

`mobility10.dat`

`mobility11.dat`

`subband_energy.dat`

`subband_velocity_x.dat`

`subband_density.dat`

The growth direction is along [001], so the carrier channel is perpendicular to this direction.
The electric field is applied in the [100] direction, so the output file `mobility10.dat`

contains the mobility into this direction.
To investigate the anisotropy in the two-dimensional plane, also the mobility calculated from the velocity in [110] direction is stored (`mobility11.dat`

).

- mobility10.dat
This file contains the calculated low field mobility, both calculated from the velocity (4th column of the file) and the diffusion constant average (5th column) obtained by the Monte Carlo procedure. The value of the electric field in [kV/cm] is given in the 1st column, and the corresponding velocity in x direction, average energy, mobility from velocity and mobility from diffusion constant are listed in the following columns.

`electric field [kV/cm] vx [cm/s] Eav [eV] µv,x [cm2/Vs] µDiff,x [cm2/Vs]`

\(\mu _{v,x} = v_x / F_x\) where \(F_x\) is the

electric fielddenoted in the 1st column.- mobility11.dat
This file contains the mobility along the [11] direction in the two-dimensional k space, and is similar to the file

`mobility10.dat`

discussed above.

`electric field [kV/cm] vxy [cm/s] Eav [eV] µv,xy [cm2/Vs]`

- subband_energy.dat
This file contains for each electric field value (1st column) the average kinetic energy within each subband (2nd column: subband 1, 3rd column: subband 2, …) in units of [eV].

`electric field [kV/cm] Eav,1 Eav,2 Eav,3 Eav,4 Eav,5 Eav,6 ...`

- subband_velocity_x.dat
This file contains the average velocity in each subband in units of [cm/s]. Its structure is similar to the previous file.

`electric field [kV/cm] vav,1 vav,2 vav,3 vav,4 vav,5 vav,6 ...`

- subband_density.dat
This file gives the occupation of each subband. \(N_{{\rm av},i}\) is the average number of particles in subband \(i\). The sum over all \(N_{{\rm av},i}\) must be equal to the number of particles that have been simulated, i.e.

`number-of-simulated-carriers`

.

`electric field [kV/cm] Nav,1 Nav,2 Nav,3 Nav,4 Nav,5 Nav,6 ...`