# 2.2.5. Mobility¶

This section describes all mobility models implemented in the nextnano Software. Related syntax can be found here.

Note

## Low-field mobility models¶

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

### Constant¶

The constant mobility model is due to lattice scattering (phonon scattering) and leads to a constant mobility that depends only on the temperature T. The lattice atoms oscillate about their equilibrium sites at finite temperature leading to a scattering of carriers which results in a temperature dependent mobility $$\mu_{const}^{n,p}$$. $$\mu_{max}^{n,p}$$ is the mobility due to bulk phonon (lattice) scattering. For all semiconductors the temperature dependent lattice mobility is modeled by a power law:

(2.2.5.1)$\mu_{const}^{n,p}(T) = \mu_{max}^{n,p} \cdot \left(\frac{T}{T_0}\right)^{-exponent},$

with temperature $$T$$ and reference temperature $$T_0 = 300K$$.

The parameter values used in this model for electrons and holes, respectively, are taken from the PhD thesis of V. Palankovski Simulation of Heterojunction Bipolar Transistors (TU Vienna). (Note: The exponent has opposite sign in his PhD thesis.)

### Masetti¶

The Masetti bulk mobility model is used to simulate the doping dependent mobility in Si and takes into account the scattering of the carriers by charged impurity ions which leads to a degradation of the carrier mobility (ionized impurity scattering). It is a model that combines lattice and impurity scattering. This model is temperature independent and the parameters are given for 300 K. Thus it is only valid for 300 K.

Following [Masetti1983], the equation for mobility is :

(2.2.5.2)$\mu^{n,p} = \mu_{min1}^{n,p} \cdot e^{ -\frac{ P_c^{n,p} }{ N_D + N_A } } + \frac{\mu_{const}^{n,p} - \mu_{min2}^{n,p}}{ 1 + \left( \frac{N_D + N_A}{C_r^{n,p}}\right)^{\alpha^{n,p}}} - \frac{\mu_{1}^{n,p}}{1 + \left(\frac{C_s^{n,p}}{N_D + N_A}\right)^{\beta^{n,p}}}$

with the reference mobility parameters $$\mu_{min1}^{n,p}$$, $$\mu_{min2}^{n,p}$$ and $$\mu_{1}^{n,p}$$, the reference doping concentration parameters $$P_c^{n,p}$$, $$C_r^{n,p}$$, $$C_s^{n,p}$$, $$\alpha^{n,p}$$ and $$\beta^{n,p}$$, and the concentration of ionized donors $$N_D$$ and acceptors $$N_A$$. The total concentration of ionized impurities is given by $$N_D + N_A$$. Note that in nextnano++ we use the nominal doping concentration as specified in the input file and not the ionized one. The low-doping reference mobility $$\mu_{const}^{n,p}$$ is determined by equation (2.2.5.1) (constant mobility-model), i.e. the values in the database under keyword mobility_constant{} are the same as under this keyword.

The values of the parameters were taken from the DESSIS documentation (2001).

### Arora¶

The Arora mobility model is used to simulate the doping dependent mobility in Si and takes into account the scattering of the carriers by charged impurity ions which leads to a degradation of the carrier mobility (ionized impurity scattering). This model is temperature dependent.

Following [Arora1982], the equation for mobility is:

(2.2.5.3)$\mu^{n,p} = \mu_{min}^{n,p}\cdot\left(\frac{T}{T_0}\right)^{\alpha_m^{n,p}} + \frac{\mu_{d}^{n,p}\cdot\left(\frac{T}{T_0}\right)^{\alpha_d^{n,p}}}{1 + \left(\frac{N_D + N_A}{N_0^{n,p}\cdot \left(\frac{T}{T_0}\right)^{\alpha_N^{n,p}}}\right)^{A_a^{n,p}\cdot\left(\frac{T}{T_0}\right)^{\alpha^{n,p}_a}}},$

with the reference mobility parameter $$\mu_{min}^{n,p}(T_0)$$, reference mobility parameter $$\mu_{d}^{n,p}$$, lattice temperature $$T$$, reference temperature $$T_0=300 K$$, reference exponent parameter $$A_a^{n,p}$$, exponents $$\alpha_N^{n,p}$$ and $$\alpha_a^{n,p}$$, reference impurity parameter $$N_0^{n,p}$$, and concentration of ionized donors $$N_D$$ and acceptors $$N_A$$. The total concentration of ionized impurities is given by $$N_A+N_D$$. Note that in nextnano++ we use the nominal doping concentration as specified in the input file and not the ionized one.

The values of the parameters were taken from the DESSIS documentation (2001).

### NINIMOS 6¶

The mobility model used in MINIMOS 6 is used to simulate the doping dependent mobility in Si and takes into account the scattering of the carriers by charged impurity ions which leads to a degradation of the carrier mobility (ionized impurity scattering). This model is temperature dependent and takes into account the reduced mobility due to lattice scattering (i.e. the values in the database under keyword mobility_constant{} are the same as under this keyword apart from the sign of the exponent). The formula of Caughey and Thomas [CaugheyThomas1967] is used together with temperature dependent coefficients. This model is well suited for Si. The equation for mobility is:

(2.2.5.4)$\mu^{n,p} = \mu_{min}^{n,p} + \frac{\mu_{const}^{n,p} - \mu_{min}^{n,p}}{1 + \left(\frac{N_D + N_A}{N_0^{n,p}\cdot \left(\frac{T}{T_0}\right)^{\alpha_N^{n,p}}}\right)^{A_a^{n,p}\cdot\left(\frac{T}{T_0}\right)^{\alpha^{n,p}_a}}},$

with lattice temperature $$T$$, reference temperature $$T_0=300 K$$, reference exponent parameter $$A_a^{n,p}$$, exponents $$\alpha_N^{n,p}$$ and $$\alpha_a^{n,p}$$, reference impurity parameter $$N_0^{n,p}$$, and concentration of ionized donors $$N_D$$ and acceptors $$N_A$$. The total concentration of ionized impurities is given by $$N_D + N_A$$. Note that in nextnano++ we use the nominal doping concentration as specified in the input file and not the ionized one. The $$\mu_{const}^{n,p}$$ is determined by the constant mobility-model: equation (2.2.5.1). The formulas for the reference mobility parameter $$\mu_{const}^{n,p}$$ are

(2.2.5.5)$\mu_{min}^{n,p}(T) = \mu_{min}^{n,p}(T_0) \left(\frac{T}{T_0}\right)^{\alpha_m^{n,p}}$
(2.2.5.6)$\mu_{min}^{n,p}(T) = \mu_{min}^{n,p}(T_0)\cdot\left(\frac{2}{3}\right)^{\alpha_m^{n,p}} \left(\frac{T}{200K}\right)^{\alpha_{m2}^{n,p}},$

where (2.2.5.5) applies to temperatures $$T \geq 200K$$ and (2.2.5.6) to temperatures $$T < 200K$$. The value $$T=200K$$ can be changed by $$T_{Switch}$$. By setting $$\alpha_m^{n,p}=\alpha_{m2}^{n,p}$$ and $$\alpha_a^{n,p} = 0$$, (2.2.5.6) reduces to (2.2.5.5) and this model can also be applied to other basic materials.

It is a model that combines lattice and impurity scattering.

The parameter values used in this model for electrons and holes, respectively, are taken from the PhD thesis of V. Palankovski Simulation of Heterojunction Bipolar Transistors (TU Vienna). (Note: The exponent has opposite sign in his PhD thesis.)

### Simba¶

Attention

These models are implemented only in nextnano³

This is one possible model for the mobility parameter $$\mu^n$$ (for electrons) and $$\mu^p$$ (for holes) that is used in the drift-diffusion model. The model is taken from the SIMBA documentation [CaugheyThomas1967]. In this model the mobility depends on the three quantities: doping density, temperature and E-field. The contributions of these quantities to the mobility are calculated in the following order:

1. Doping concentration:

(2.2.5.7)$\mu^{n,p}(N_A+N_D) = \mu_{min}^{n,p} + \frac{\mu_{D}^{n,p}}{ 1 + \left(\frac{N_A + N_D}{N_{ref}^{n,p}} \right)^{\alpha^{n,p}}}$

with minimum mobility $$\mu_{min}^{n,p}$$, reference doping density $$N_{ref}^{n,p}$$, reference mobility $$\mu_{D}^{n,p}$$, exponent $$\alpha^{n,p}$$ and concentration of ionized acceptors $$N_A$$ and donors $$N_D$$. Note that in nextnano++ we use the nominal doping concentration as specified in the input file and not the ionized one.

2. Temperature:

(2.2.5.8)$\mu^{n,p}(T) = \mu_{max}^{n,p} \cdot \left(\frac{T}{T_0}\right)^{-\gamma^{n,p}},$

with temperature $$T$$, reference temperature $$T_0$$ and exponent for temperature dependence $$\gamma^{n,p}$$.

3. Electric field (perpendicular):

(2.2.5.9)$\mu^{n,p}(\mathbf{E_\perp}) = \frac{\mu^{n,p}}{\sqrt{1 + \frac{\|\mathbf{E_\perp}\|}{E_T^{n,p}}}}$

with perpendicular electric field parameter $$E_T^{n,p}$$. It is possible to include/ exclude the perpendicular E-field dependence.

4. Electric field (parallel):

There are six different SIMBA models for including the impact of the parallel electric field:

Model 0

no dependence on parallel electric field

Model 1

(2.2.5.10)$\mu^{n,p}(\mathbf{E}) = \frac{\mu^{n,p}}{\left[1+\left(\frac{\|\mathbf{E}\|}{E_p^{n,p}} \right)^{\alpha^{n,p}}\right]^{\beta^{n,p}}}$

with exponents $$\alpha^{n,p}$$ and $$\beta^{n,p}$$. The temperature dependency of peak electric field is described by:

(2.2.5.11)$E_p^{n,p}(T) = E_0^{n,p} - d_E^{n,p} \cdot (T-T_0),$

with temperature $$T$$, peak electric field $$E_0^{n,p}$$, temperature dependence parameter of peak electric field $$d_E^{n,p}$$ and reference temperature $$T_0$$.

Model 2

(2.2.5.12)$\mu^{n,p}(\mathbf{E}) = \frac{\mu^{n,p}}{\left[1+\left(\mu^{n,p}\frac{\|\mathbf{E}\|}{v_s^{n,p}} \right)^{\kappa^{n,p}}\right]^{1/\kappa^{n,p}}}$

with exponent $$\kappa^{n,p}$$ adn saturation velocity v_s^{n,p}(T). Temperature dependency of saturation velocity is described by:

(2.2.5.13)$v_s^{n,p}(T) = v_0^{n,p} - d_v^{n,p} \cdot (T-T_0),$

with reference saturation velocity $$v_0^{n,p}$$, temperature dependence parameter of saturation velocity $$d_v^{n,p}$$ and reference Temperature $$T_0^{n,p}$$.

Model 3

(2.2.5.14)$\mu^{n,p}(\mathbf{E}) = \frac{\mu^{n,p} + v_s^{n,p}\frac{\|\mathbf{E}\|^3}{E_p^{n,p}}}{\left[1+\left(\frac{\|\mathbf{E}\|}{\left(E_p^{n,p}\right)^{4}} \right)^{\alpha^{n,p}}\right]^{\beta^{n,p}}}$

Model 4

(2.2.5.15)$\mu^{n,p}(\mathbf{E}) = \frac{2\mu^{n,p}}{1 + \left[1+\left(\mu^{n,p}\frac{\|\mathbf{E}\|}{v_s^{n,p}} \right)^{\kappa^{n,p}}\right]^{1/\kappa^{n,p}}}$

Model 5

(2.2.5.16)$\mu^{n,p}(\mathbf{E}) = \frac{\mu^{n,p} + v_s^{n,p}\frac{\left(\|\mathbf{E}\|\right)^{\alpha^{n,p}-1}}{\left(E_p^{n,p}\right)^{\alpha^{n,p}}}}{\left[1+\left(\frac{\|\mathbf{E}\|}{E_p^{n,p}} \right)^{\alpha^{n,p}}\right]^{\beta^{n,p}}}$

## High-Field Mobility Models¶

Four high-field mobility models are currently implemented in nextnano++. In our implementation, each of them uses results obtained from selected low-field model passed via $$\mu _\text{low}$$.

### Hänsch¶

As mentioned above, this model is a special case of the Extended Canali model in the limit of strong surface scattering defined by W. Hänch and M. Miura-Mattausch

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

where $$\mu _\text{low}$$ is low-field mobility, $$v_ \text{sat}$$ is saturation velocity, and $$F$$ is the driving force.

### Extended Canali¶

The Extended Canali model is an extended version of Jacoboni-Canali model, originally applied to electron and hole drift-velocity measurements in silicon by Canali, et al..

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

where $$\mu _\text{low}$$ is low-field mobility, $$v_ \text{sat}$$ is saturation velocity, and $$F$$ is the driving force. Parameters $$\alpha$$, $$\beta$$ and $$v_ \text{sat}$$ are defined independently for holes and electrons . The driving force $$F$$ of the respective carriers is evaluated as the gradient of the respective quasi-Fermi level. The $$\alpha$$ parameter should be set to zero, if one aims at using the Extended Canali model. One can transform it into Hänch model by setting $$\alpha=1$$ and $$\beta=2$$.

### Transferred-Electron¶

The transferred electron model below bases on Monte Carlo simulation of transport in the III-nitride wurtzite materials done by M. Farahmand, et al..

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

where $$\mu _\text{low}$$ is low-field mobility, $$v_ \text{sat}$$ is saturation velocity, $$F$$ is the driving force, and $$E_0$$ is critical field. Parameters $$\alpha$$, $$\beta$$, $$\gamma$$ and $$v_ \text{sat}$$ are defined independently for holes and electrons.

### Eastman-Tiwari-Shur¶

A model based on a modified theory of the high-field domains which takes into account the field dependent diffusion by L. F. Eastman, et al. for GaAs MESFETs. Where $$E_s \equiv \frac{v_{sat}}{{\mu_{low}}}$$ after work of J. Chillieri, et al..

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

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

Parameters $$\alpha$$ and $$\beta$$ can be replaced introducing four other parameters $$E_\text{peak}$$, $$E_\text{mid}$$, $$v_\text{peak}$$, and $$v_\text{mid}$$, all related to the shape of the drift velocity function of the driving force. See J. Chillieri, et al. for reference.

$\beta = \frac{ log\left( \frac{E_\text{mid}\mu_\text{low}-v_\text{mid}}{E_\text{peak}\mu_\text{low}-v_\text{peak}} \cdot \frac{v_\text{peak}-v_\text{sat}}{v_\text{mid}-v_\text{sat}} \right)} {log\left(\frac{E_\text{mid}}{E_\text{peak}}\right)}$
$\alpha = \frac{E_\text{peak}\mu_\text{low}-v_\text{peak}}{v_\text{peak}-v_\text{sat}}\left(\frac{v_\text{sat}}{E_\text{peak}\mu_\text{low}}\right)^\beta$