Mobility Models in Nuwa TCAD

Mobility in semiconductors refers to the ability of charge carriers, such as electrons and holes, to move through the material when an electric field is applied. This property directly influences the conductivity and overall performance of semiconductor devices. Mobility models are mathematical representations that describe how mobility varies under different conditions, such as electric fields, temperature, and doping levels. In silicon (Si) semiconductors, both high-field and low-field mobility models are essential for accurate simulation and design of electronic devices. High-field models account for carrier velocity saturation at strong electric fields, while low-field models describe the linear relationship between mobility and electric field at weaker fields. These models are vital in enabling precise predictions of device behavior and optimizing the device performance. In today’s article, we will introduce the mobility models of electrons and holes in silicon (Si) materials, covering both high-field and low-field scenarios.

1. Electron/Hole Low-Field Mobility Model

Low-field mobility models describe the behavior of charge carriers when the applied electric field is relatively weak. In these conditions, the mobility of electrons and holes typically follows a linear relationship with the electric field, allowing for straightforward predictions of carrier movement.

1.1 Analytic Low-Field Mobility Model

This model is used to calculate ${\mu }_0\mu \_\{0\}$.

The following analytic function based upon the work of Caughey and Thomas[1,2] can be used to specify doping- and temperature-dependent low-field mobilities.

$${\mu }_0={\mu }_1\cdot (\frac{T_L}{300}{)}^{\alpha }+\frac{{\mu }_2\cdot (\frac{T_L}{300}{)}^{\beta }-{\mu }_1\cdot (\frac{T_L}{300}{)}^{\alpha }}{1+(\frac{N}{N_{crit}}{)}^{\delta }\cdot (\frac{T_L}{300}{)}^{\gamma }}\backslash mu\_\{0\}\; =\; \backslash mu\_\{1\}\backslash cdot(\backslash frac\{T\_L\}\{300\})^\{\backslash alpha\}+\backslash frac\{\backslash mu\_\{2\}\; \backslash cdot\; (\backslash frac\{T\_L\}\{300\})^\{\backslash beta\}\; -\; \backslash mu\_\{1\}\backslash cdot(\backslash frac\{T\_L\}\{300\})^\{\backslash alpha\}\}\{1\; +\; (\backslash frac\{N\}\{N\_\{crit\}\})^\{\backslash delta\}\; \backslash cdot\; (\backslash frac\{T\_L\}\{300\})^\{\backslash gamma\}\}$$

where $NN$ is the total local dopant concentration. The parameters for this model with their default values (Arsenic-doped Silicon for electron values and Boron-doped Silicon for hole values) are defined in the following table:

Symbol	Parameter Name	Electron Value	Hole Value	Units
μ1	mu1	0.005524	0.00497	m2/(V*s)
μ2	mu2	0.142923	0.047937	m2/(V*s)
α	alpha	0.0	0.0	N/A
β	beta	-2.3	-2.2	N/A
γ	gamma	-3.8	-3.7	N/A
𝛿	delta	0.73	0.70	N/A
Ncrit	Ncrit	1.072×1023	1.606×1023	m-3

The type of corresponding parameters are listed in the following table:

Symbol	Updated During Simulation	Predefinition of Material Parameter	Device Configuration	Description
μ1	✗	✓	✗	N/A
μ2	✗	✓	✗	N/A
α	✗	✓	✗	N/A
β	✗	✓	✗	N/A
γ	✗	✓	✗	N/A
𝛿	✗	✓	✗	N/A
Ncrit	✗	✓	✗	N/A
N	✗	✗	✓	Total concentration (m-3)
TL	✗	✗	✓	Lattice temperature (K)

Reference:

[1] Caughey, D. Mo, and R. E. Thomas. "Carrier mobilities in silicon empirically related to doping and field." Proceedings of the IEEE 55.12 (1967): 2192-2193. doi: 10.1109/PROC.1967.6123.

[2] Pinto, Mark R., Conor S. Rafferty, and Robert W. Dutton. PISCES II: Poisson and continuity equation solver. 1984.

1.2 Analytic Low-Field Mobility Model Ⅱ

This model is used to calculate ${\mu }_0\mu \_\{0\}$.

The following analytic function based upon the work of Caughey and Thomas[1,2] can be used to specify doping- and temperature-dependent low-field mobilities.

$${\mu }_0={\mu }_1\cdot (\frac{T_L}{300}{)}^{\alpha }+\frac{{\mu }_2\cdot (\frac{T_L}{300}{)}^{\beta }-{\mu }_1\cdot (\frac{T_L}{300}{)}^{\alpha }}{1+(\frac{N}{N_{crit}\cdot (\frac{T_L}{300}{)}^{\gamma }}{)}^{\delta \cdot (\frac{T_L}{300}{)}^{\eta }}}\backslash mu\_\{0\}\; =\; \backslash mu\_\{1\}\backslash cdot(\backslash frac\{T\_L\}\{300\})^\{\backslash alpha\}+\backslash frac\{\backslash mu\_\{2\}\; \backslash cdot\; (\backslash frac\{T\_L\}\{300\})^\{\backslash beta\}\; -\; \backslash mu\_\{1\}\backslash cdot(\backslash frac\{T\_L\}\{300\})^\{\backslash alpha\}\}\{1\; +\; (\backslash frac\{N\}\{N\_\{crit\}\backslash cdot(\backslash frac\{T\_L\}\{300\})^\{\backslash gamma\}\})^\{\backslash delta\; \backslash cdot(\backslash frac\{T\_L\}\{300\})^\{\backslash eta\}\}\}$$

where $NN$ is the total local dopant concentration. The parameters for this model with their default values (n-type or p-type 6H-SiC[3]) are defined in the following table:

Symbol	Parameter Name	Electron Value	Hole Value	Units
μ1	mu1	0.003	0.001	m2/(V*s)
μ2	mu2	0.042	0.008	m2/(V*s)
α	alpha	-0.5	-0.5	N/A
β	beta	-2.5	-2.15	N/A
γ	gamma	2.5	2.23	N/A
𝛿	delta	0.8	0.34	N/A
η	eta	0.5	0.0	N/A
Ncrit	Ncrit	6×1023	1.76×1025	m-3

The type of corresponding parameters are listed in the following table:

Symbol	Updated During Simulation	Predefinition of Material Parameter	Device Configuration	Description
μ1	✗	✓	✗	N/A
μ2	✗	✓	✗	N/A
α	✗	✓	✗	N/A
β	✗	✓	✗	N/A
γ	✗	✓	✗	N/A
𝛿	✗	✓	✗	N/A
η	✗	✓	✗	N/A
Ncrit	✗	✓	✗	N/A
N	✗	✗	✓	Total concentration (m-3)
TL	✗	✗	✓	Lattice temperature (K)

Reference:

[1] Caughey, D. Mo, and R. E. Thomas. "Carrier mobilities in silicon empirically related to doping and field." Proceedings of the IEEE 55.12 (1967): 2192-2193. doi: 10.1109/PROC.1967.6123.

[2] Pinto, Mark R., Conor S. Rafferty, and Robert W. Dutton. PISCES II: Poisson and continuity equation solver. 1984.

[3] Roschke, Matthias, and Frank Schwierz. "Electron mobility models for 4H, 6H, and 3C SiC [MESFETs]." IEEE Transactions on electron devices 48.7 (2001): 1442-1447. doi:10.1109/16.930664.

1.3 Arora Model

Arora model is used to calculate ${\mu }_0\mu \_\{0\}$.

Arora model[1] includes the effect of dopping and temperature.This model has the following form:

$${\mu }_0={\mu }_1\cdot (\frac{T_L}{300}{)}^{\alpha }+\frac{{\mu }_2\cdot (\frac{T_L}{300}{)}^{\beta }}{1+\frac{N}{N_{crit}\cdot (\frac{T_L}{300}{)}^{\gamma }}}\backslash mu\_\{0\}\; =\; \backslash mu\_\{1\}\backslash cdot\; (\backslash frac\{T\_L\}\{300\})^\{\backslash alpha\}+\backslash frac\{\backslash mu\_\{2\}\backslash cdot(\backslash frac\{T\_L\}\{300\})^\{\backslash beta\}\}\{1+\backslash frac\{N\}\{N\_\{crit\}\backslash cdot(\backslash frac\{T\_L\}\{300\})^\{\backslash gamma\}\}\}$$

where $NN$ is the total local dopant concentration.

The parameters for this model with their default values (Phosphorus-doped Silicon for electron values and Boron-doped Silicon for hole values) are defined in the following table:

Symbol	Parameter Name	Electron Value	Hole Value	Units
μ1	mu1	0.00880	0.00543	m2/(V*s)
μ2	mu2	0.1252	0.0407	m2/(V*s)
α	alpha	-0.57	-0.57	N/A
β	beta	-2.33	-2.33	N/A
γ	gamma	2.546	2.546	N/A
Ncrit	Ncrit	1.432×1023	2.67×1023	m-3

The type of corresponding parameters are listed in the following table:

Symbol	Updated During Simulation	Predefinition of Material Parameter	Device Configuration	Description
μ1	✗	✓	✗	N/A
μ2	✗	✓	✗	N/A
α	✗	✓	✗	N/A
β	✗	✓	✗	N/A
γ	✗	✓	✗	N/A
Ncrit	✗	✓	✗	N/A
N	✗	✗	✓	Total concentration (m-3)
TL	✗	✗	✓	Lattice temperature (K)

Reference:

Arora, Narain D., John R. Hauser, and David J. Roulston. "Electron and hole mobilities in silicon as a function of concentration and temperature." IEEE Transactions on electron devices 29.2 (1982): 292-295. doi: 10.1109/T-ED.1982.20698.

1.4 University of Bologna (bulk)

This model is used to calculate ${\mu }_0\mu \_\{0\}$.

This model implements the following relationship[1]:

$${\mu }_0={\mu }_1+\frac{{\mu }_L-{\mu }_1}{1+{\left(\frac{N_D}{C_{r1}{\left(\frac{T_L}{300}\right)}^{{\gamma }_{r1}}}\right)}^{\alpha }+{\left(\frac{N_A}{C_{r2}{\left(\frac{T_L}{300}\right)}^{{\gamma }_{r2}}}\right)}^{\beta }}-\frac{{\mu }_2}{1+{(\frac{N_D}{C_{s1}{\left(\frac{T_L}{300}\right)}^{{\gamma }_{s1}}}+\frac{N_A}{C_{s2}})}^{-2}}\backslash mu\_\{0\}=\backslash mu\_\{1\}+\backslash frac\{\backslash mu\_\{L\}-\backslash mu\_\{1\}\}\{1+\backslash left(\backslash frac\{N\_\{D\}\}\{C\_\{r\; 1\}\; \backslash left(\backslash frac\{T\_L\}\{300\}\backslash right)^\{\backslash gamma\_\{r\; 1\}\}\}\backslash right)^\{\backslash alpha\}+\backslash left(\backslash frac\{N\_\{A\}\}\{C\_\{r\; 2\}\; \backslash left(\backslash frac\{T\_L\}\{300\}\backslash right)^\{\backslash gamma\_\{r\; 2\}\}\}\backslash right)^\{\backslash beta\}\}-\backslash frac\{\backslash mu\_\{2\}\}\{1+\backslash left(\backslash frac\{N\_\{D\}\}\{C\_\{s\; 1\}\; \backslash left(\backslash frac\{T\_L\}\{300\}\backslash right)^\{\backslash gamma\_\{s\; 1\}\}\}+\backslash frac\{N\_\{A\}\}\{C\_\{s\; 2\}\}\backslash right)^\{-2\}\}$$

where,

$${\mu }_L={\mu }_{\max }{\left(\frac{T_L}{300}\right)}^{-\gamma +c\left(\frac{T_L}{300}\right)}\backslash mu\_\{L\}=\backslash mu\_\{\backslash max\; \}\; \backslash left(\backslash frac\{T\_L\}\{300\}\backslash right)^\{-\backslash gamma+c\; \backslash left(\backslash frac\{T\_L\}\{300\}\backslash right)\}$$ $${\mu }_1=\frac{{\mu }_{0d}{\left(\frac{T_L}{300}\right)}^{-{\gamma }_{0d}}{N}_D+{\mu }_{0a}{\left(\frac{T_L}{300}\right)}^{-{\gamma }_{0a}}{N}_A}{N_A+N_D}\backslash mu\_\{1\}=\backslash frac\{\backslash mu\_\{0\; d\}\; \backslash left(\backslash frac\{T\_L\}\{300\}\backslash right)^\{-\backslash gamma\_\{0\; d\}\}\; N\_\{D\}+\backslash mu\_\{0\; a\}\; \backslash left(\backslash frac\{T\_L\}\{300\}\backslash right)^\{-\backslash gamma\_\{0\; a\}\}\; N\_\{A\}\}\{N\_\{A\}+N\_\{D\}\}$$ $${\mu }_2=\frac{{\mu }_{1d}{\left(\frac{T_L}{300}\right)}^{-{\gamma }_{1d}}{N}_D+{\mu }_{1a}{\left(\frac{T_L}{300}\right)}^{-{\gamma }_{1a}}{N}_A}{N_A+N_D}\backslash mu\_\{2\}=\backslash frac\{\backslash mu\_\{1\; d\}\; \backslash left(\backslash frac\{T\_L\}\{300\}\backslash right)^\{-\backslash gamma\_\{1\; d\}\}\; N\_\{D\}+\backslash mu\_\{1\; a\}\; \backslash left(\backslash frac\{T\_L\}\{300\}\backslash right)^\{-\backslash gamma\_\{1\; a\}\}\; N\_\{A\}\}\{N\_\{A\}+N\_\{D\}\}$$

where $N_{D}N\_\{D\}$ is the donor concentration in $m^{-3}m^\{-3\}$ , $N_{A}N\_\{A\}$ is the acceptor concentration in $m^{-3}m^\{-3\}$.

The parameters for this model with their default values are defined in the following table:

Symbol	Parameter Name	Electron Value	Hole Value	Units
μmax	mumax	0.1441	0.04705	m2/(V*s)
c	c	0.07	0.0	N/A
γ	gamma	2.45	2.16	N/A
γ0d	gamm0d	0.6	1.3	N/A
μ0d	mu0d	0.0055	0.0090	m2/(V*s)
γ0a	gamma0a	1.3	0.7	N/A
μ0a	mu0a	0.0132	0.0044	m2/(V*s)
γ1d	gamma1d	0.5	2.0	N/A
μ1d	mu1d	0.00424	0.00282	m2/(V*s)
γ1a	gamma1a	1.25	0.8	N/A
μ1a	mu1a	0.00735	0.00282	m2/(V*s)
γr1	gammar1	3.65	2.2	N/A
Cr1	Cr1	8.9×1022	1.3×1024	m-3
γr2	gammar2	2.65	3.1	N/A
Cr2	Cr2	1.22×1023	2.45×1023	m-3
γs1	gammas1	0.0	6.2	N/A
Cs1	Cs1	2.9×1026	1.1×1024	m-3
Cs2	Cs2	7×1026	6.1×1026	m-3
α	alpha	0.68	0.77	N/A
β	beta	0.72	0.719	N/A

The type of corresponding parameters are listed in the following table:

Symbol	Updated During Simulation	Predefinition of Material Parameter	Device Configuration	Description
μmax	✗	✓	✗	N/A
c	✗	✓	✗	N/A
γ	✗	✓	✗	N/A
γ0d	✗	✓	✗	N/A
μ0d	✗	✓	✗	N/A
γ0a	✗	✓	✗	N/A
μ0a	✗	✓	✗	N/A
γ1d	✗	✓	✗	N/A
μ1d	✗	✓	✗	N/A
γ1a	✗	✓	✗	N/A
μ1a	✗	✓	✗	N/A
γr1	✗	✓	✗	N/A
Cr1	✗	✓	✗	N/A
γr2	✗	✓	✗	N/A
Cr2	✗	✓	✗	N/A
γs1	✗	✓	✗	N/A
Cs1	✗	✓	✗	N/A
Cs2	✗	✓	✗	N/A
α	✗	✓	✗	N/A
β	✗	✓	✗	N/A
NA	✗	✗	✓	Acceptor concentration (m-3)
ND	✗	✗	✓	Donor concentration (m-3)
TL	✗	✗	✓	Lattice temperature (K)

Reference:

[1] Reggiani, S., et al. "A unified analytical model for bulk and surface mobility in Si n-and p-channel MOSFET's." 29th European solid-state device research conference. Vol. 1. IEEE, 1999.

1.5 University of Bologna Inversion Layer

This model is used to calculate ${\mu }_0\mu \_\{0\}$.

This model combines the scattering efects from Coulombic effects (${\mu }_{cb}\mu \_\{cb\}$), surface acoustic phonons (${\mu }_{ac}\mu \_\{ac\}$) and surface roughness (${\mu }_{rough}\mu \_\{rough\}$) using the following relationship[1]:

$$\frac{1}{{\mu }_0}=\frac{1}{{\mu }_{cb}}+\frac{D}{{\mu }_{ac}}+\frac{D}{{\mu }_{rough}}\backslash frac\{1\}\{\backslash mu\_\{0\}\}\; =\; \backslash frac\{1\}\{\backslash mu\_\{cb\}\}\; +\; \backslash frac\{D\}\{\backslash mu\_\{ac\}\}\; +\; \backslash frac\{D\}\{\backslash mu\_\{rough\}\}$$

where,

$$D=exp(-\frac{l}{l_{crit}})D\; =\; exp(-\backslash frac\{l\}\{l\_\{crit\}\})$$

where $ll$ is the distance to the nearest interface from the point where the mobility will be calculated. The Coulomb term and screening effects are given by:

$${\mu }_{cb}={\mu }_{bulk}[D{(1+f_{sc}^{\tau })}^{\frac{1}{\tau }}+(1-D)]\backslash mu\_\{c\; b\}=\backslash mu\_\{b\; u\; l\; k\}\backslash left[D\backslash left(1+f\_\{s\; c\}^\{\backslash tau\}\backslash right)^\{\backslash frac\{1\}\{\backslash tau\}\}+(1-D)\backslash right]$$ $$f_{sc}={\left(\frac{N_1}{N_A+N_D}\right)}^{\eta }\frac{N_{\min }}{N_A+N_D}f\_\{s\; c\}=\backslash left(\backslash frac\{N\_\{1\}\}\{N\_\{A\}+N\_\{D\}\}\backslash right)^\{\backslash eta\}\; \backslash frac\{N\_\{\backslash min\; \}\}\{N\_\{A\}+N\_\{D\}\}$$

where ${\mu }_{bulk}\mu \_\{bulk\}$ can get from University of Bologna Bulk Model, $N_{min}N\_\{min\}$ is the minority carrier concentration, $N_{D}N\_\{D\}$ is the donor concentration in $m^{-3}m^\{-3\}$, $N_{A}N\_\{A\}$ is the acceptor concentration in $m^{-3}m^\{-3\}$.

${\mu }_{bulk}\mu \_\{bulk\}$ is given by[1]:

$${\mu }_{bulk}={\mu }_1+\frac{{\mu }_L-{\mu }_1}{1+{\left(\frac{N_D}{C_{r1}{\left(\frac{T_L}{300}\right)}^{{\gamma }_{r1}}}\right)}^{\alpha }+{\left(\frac{N_A}{C_{r2}{\left(\frac{T_L}{300}\right)}^{{\gamma }_{r2}}}\right)}^{\beta }}-\frac{{\mu }_2}{1+{(\frac{N_D}{C_{s1}{\left(\frac{T_L}{300}\right)}^{{\gamma }_{s1}}}+\frac{N_A}{C_{s2}})}^{-2}}\backslash mu\_\{bulk\}=\backslash mu\_\{1\}+\backslash frac\{\backslash mu\_\{L\}-\backslash mu\_\{1\}\}\{1+\backslash left(\backslash frac\{N\_\{D\}\}\{C\_\{r\; 1\}\; \backslash left(\backslash frac\{T\_L\}\{300\}\backslash right)^\{\backslash gamma\_\{r\; 1\}\}\}\backslash right)^\{\backslash alpha\}+\backslash left(\backslash frac\{N\_\{A\}\}\{C\_\{r\; 2\}\; \backslash left(\backslash frac\{T\_L\}\{300\}\backslash right)^\{\backslash gamma\_\{r\; 2\}\}\}\backslash right)^\{\backslash beta\}\}-\backslash frac\{\backslash mu\_\{2\}\}\{1+\backslash left(\backslash frac\{N\_\{D\}\}\{C\_\{s\; 1\}\; \backslash left(\backslash frac\{T\_L\}\{300\}\backslash right)^\{\backslash gamma\_\{s\; 1\}\}\}+\backslash frac\{N\_\{A\}\}\{C\_\{s\; 2\}\}\backslash right)^\{-2\}\}$$

where,

$${\mu }_L={\mu }_{\max }{\left(\frac{T_L}{300}\right)}^{-{\gamma }_L+c\left(\frac{T_L}{300}\right)}\backslash mu\_\{L\}=\backslash mu\_\{\backslash max\; \}\; \backslash left(\backslash frac\{T\_L\}\{300\}\backslash right)^\{-\backslash gamma\_L+c\; \backslash left(\backslash frac\{T\_L\}\{300\}\backslash right)\}$$ $${\mu }_1=\frac{{\mu }_{0d}{\left(\frac{T_L}{300}\right)}^{-{\gamma }_{0d}}{N}_D+{\mu }_{0a}{\left(\frac{T_L}{300}\right)}^{-{\gamma }_{0a}}{N}_A}{N_A+N_D}\backslash mu\_\{1\}=\backslash frac\{\backslash mu\_\{0\; d\}\; \backslash left(\backslash frac\{T\_L\}\{300\}\backslash right)^\{-\backslash gamma\_\{0\; d\}\}\; N\_\{D\}+\backslash mu\_\{0\; a\}\; \backslash left(\backslash frac\{T\_L\}\{300\}\backslash right)^\{-\backslash gamma\_\{0\; a\}\}\; N\_\{A\}\}\{N\_\{A\}+N\_\{D\}\}$$ $${\mu }_2=\frac{{\mu }_{1d}{\left(\frac{T_L}{300}\right)}^{-{\gamma }_{1d}}{N}_D+{\mu }_{1a}{\left(\frac{T_L}{300}\right)}^{-{\gamma }_{1a}}{N}_A}{N_A+N_D}\backslash mu\_\{2\}=\backslash frac\{\backslash mu\_\{1\; d\}\; \backslash left(\backslash frac\{T\_L\}\{300\}\backslash right)^\{-\backslash gamma\_\{1\; d\}\}\; N\_\{D\}+\backslash mu\_\{1\; a\}\; \backslash left(\backslash frac\{T\_L\}\{300\}\backslash right)^\{-\backslash gamma\_\{1\; a\}\}\; N\_\{A\}\}\{N\_\{A\}+N\_\{D\}\}$$

where $N_{D}N\_\{D\}$ is the donor concentration in $m^{-3}m^\{-3\}$, $N_{A}N\_\{A\}$ is the acceptor concentration in $m^{-3}m^\{-3\}$.

The surface scattering terms are defined as:

$${\mu }_{ac}=c{\left(\frac{T_L}{300}\right)}^{-{\gamma }_c}{\left(\frac{N_A+N_D}{N_2}\right)}^a{F}_{\perp }^{-\delta }\backslash mu\_\{a\; c\}\; =c\backslash left(\backslash frac\{T\_L\}\{300\}\backslash right)^\{-\backslash gamma\_\{c\}\}\backslash left(\backslash frac\{N\_\{A\}+N\_\{D\}\}\{N\_\{2\}\}\backslash right)^\{a\}\; F\_\{\backslash perp\}^\{-\backslash delta\}$$ $${\mu }_{\text{rough }}=d{\left(\frac{T_L}{300}\right)}^{{\gamma }_d}{\left(\frac{N_A+N_D+N_3}{N_4}\right)}^b{F}_{\perp }^{-\gamma }\backslash mu\_\{\backslash text\; \{rough\; \}\}\; =d\backslash left(\backslash frac\{T\_L\}\{300\}\backslash right)^\{\backslash gamma\_\{d\}\}\backslash left(\backslash frac\{N\_\{A\}+N\_\{D\}+N\_\{3\}\}\{N\_\{4\}\}\backslash right)^\{b\}\; F\_\{\backslash perp\}^\{-\backslash gamma\}$$

where $F_{\text{⊥}}F\_\{\perp \}$ is the electric field components perpendicular to the current density vector and $T_{L}T\_\{L\}$ is lattice temperature.

The parameters for this model with their default values (n-MOSFET for electron values and p-MOSFET for hole values) are defined in the following table:

Symbol	Parameter Name	Electron Value	Hole Value	Units
μmax	mumax	0.1441	0.04705	m2/(V*s)
c	c	0.07	0.0	N/A
γL	gammaL	2.45	2.16	N/A
γ0d	gamm0d	0.6	1.3	N/A
μ0d	mu0d	0.0055	0.0090	m2/(V*s)
γ0a	gamma0a	1.3	0.7	N/A
μ0a	mu0a	0.0132	0.0044	m2/(V*s)
γ1d	gamma1d	0.5	2.0	N/A
μ1d	mu1d	0.00424	0.00282	m2/(V*s)
γ1a	gamma1a	1.25	0.8	N/A
μ1a	mu1a	0.00735	0.00282	m2/(V*s)
γr1	gammar1	3.65	2.2	N/A
Cr1	Cr1	8.9×1022	1.3×1024	m-3
γr2	gammar2	2.65	3.1	N/A
Cr2	Cr2	1.22×1023	2.45×1023	m-3
γs1	gammas1	0.0	6.2	N/A
Cs1	Cs1	2.9×1026	1.1×1024	m-3
Cs2	Cs2	7×1026	6.1×1026	m-3
α	alpha	0.68	0.77	N/A
β	beta	0.72	0.719	N/A
N1	N1	2.34×1022	2.02×1022	m-3
N2	N2	4×1021	7.8×1021	m-3
N3	N3	1.0×1023	2×1021	m-3
N4	N4	2.4×1024	6.6×1023	m-3
c	c	1.8	0.5726	m2/(V*s)
γc	gammac	1.6	1.3	N/A
d	d	5.8×1014	7.82×1011	m2/(V*s)
γd	gammad	0	1.4	N/A
τ	tau	1.0	3.0	N/A
η	eta	0.3	0.5	N/A
a	a	0.026	-0.02	N/A
b	b	0.11	0.08	N/A
lcrit	lcrit	10-8	10-8	m
δ	delta	0.29	0.3	N/A
γ	gamma	2.64	2.24	N/A

The type of corresponding parameters are listed in the following table:

Symbol	Updated During Simulation	Predefinition of Material Parameter	Device Configuration	Description
F⊥	✓	✗	✗	Transverse electric field (V/m)
l	✓	✗	✗	The distance to the nearest interface from the simulation point (m)
μmax	✗	✓	✗	N/A
c	✗	✓	✗	N/A
γL	✗	✓	✗	N/A
γ0d	✗	✓	✗	N/A
μ0d	✗	✓	✗	N/A
γ0a	✗	✓	✗	N/A
μ0a	✗	✓	✗	N/A
γ1d	✗	✓	✗	N/A
μ1d	✗	✓	✗	N/A
γ1a	✗	✓	✗	N/A
μ1a	✗	✓	✗	N/A
γr1	✗	✓	✗	N/A
Cr1	✗	✓	✗	N/A
γr2	✗	✓	✗	N/A
Cr2	✗	✓	✗	N/A
γs1	✗	✓	✗	N/A
Cs1	✗	✓	✗	N/A
Cs2	✗	✓	✗	N/A
α	✗	✓	✗	N/A
β	✗	✓	✗	N/A
N1	✗	✓	✗	N/A
N2	✗	✓	✗	N/A
N3	✗	✓	✗	N/A
N4	✗	✓	✗	N/A
c	✗	✓	✗	N/A
γc	✗	✓	✗	N/A
d	✗	✓	✗	N/A
γd	✗	✓	✗	N/A
τ	✗	✓	✗	N/A
η	✗	✓	✗	N/A
a	✗	✓	✗	N/A
b	✗	✓	✗	N/A
δ	✗	✓	✗	N/A
γ	✗	✓	✗	N/A
lcrit	✗	✗	✓	N/A
NA	✗	✗	✓	Acceptor concentration (m-3)
ND	✗	✗	✓	Donor concentration (m-3)
TL	✗	✗	✓	Lattice temperature (K)

Issues:

The default parameters fit very low doping concentration well, but those parameters still need to fit with a higher doping concentration (Fig. 1 in ref 1)

Reference:

Reggiani, S., et al. "A unified analytical model for bulk and surface mobility in Si n-and p-channel MOSFET's." 29th European solid-state device research conference. Vol. 1. IEEE, 1999.

1.6 Dorkel-Leturcq Model

This model is used to calculate ${\mu }_0\backslash mu\_\{0\}$.

The Dorkel and Leturcq Model^[1] for low-field mobility includes the dependence on temperature, doping, and carrier-carrier scattering. This model has the form:

$${\mu }_{n_0,p_0}={\mu }_{n,p}^L[\left(\frac{1.025}{1+{\left[2.126\left(\frac{{\mu }_{n,p}^L}{{\mu }_{n,p}^{IC}}\right)\right]}^{0.715}}\right)-0.025]\backslash mu\_\{n\_0,\; p\_0\}=\backslash mu\_\{n,\; p\}^\{L\}\backslash left[\backslash left(\backslash frac\{1.025\}\{1+\backslash left[2.126\backslash left(\backslash frac\{\backslash mu\_\{n,\; p\}^\{L\}\}\{\backslash mu\_\{n,\; p\}^\{IC\}\}\backslash right)\backslash right]^\{0.715\}\}\backslash right)-0.025\backslash right]$$

where ${\mu }^L\backslash mu^L$ is the lattice scattering, ${\mu }^I\backslash mu^I$ is the ionized impurity scattering, and ${\mu }^C\backslash mu^C$ is the carrier-carrier scattering. Here, ${\mu }^{IC}\backslash mu^\{IC\}$ is defined as:

$${\mu }_{n,p}^{IC}={[\frac{1}{{\mu }^C}+\frac{1}{{\mu }_{n,p}^I}]}^{-1}\backslash mu\_\{n,\; p\}^\{I\; C\}=\backslash left[\backslash frac\{1\}\{\backslash mu^\{C\}\}+\backslash frac\{1\}\{\backslash mu\_\{n,\; p\}^\{I\}\}\backslash right]^\{-1\}$$

where

$${\mu }^C=\frac{1.04\cdot 10^{21}{\left(\frac{T_L}{300}\right)}^{3\mathrm{/}2}}{\sqrt{np}\ln [1+7.45\cdot 10^{13}{\left(\frac{T_L}{300}\right)}^2(np{)}^{-1\mathrm{/}3}]}\backslash mu^\{C\}=\backslash frac\{1.04\; \backslash cdot\; 10^\{21\}\backslash left(\backslash frac\{T\_\{L\}\}\{300\}\backslash right)^\{3\; /\; 2\}\}\{\backslash sqrt\{n\; p\}\; \backslash ln\; \backslash left[1+7.45\; \backslash cdot\; 10^\{13\}\backslash left(\backslash frac\{T\_\{L\}\}\{300\}\backslash right)^\{2\}(n\; p)^\{-1\; /\; 3\}\backslash right]\}$$ $${\mu }_n^I=\frac{A_N\cdot {\left(T_L\right)}^{3\mathrm{/}2}}{N}f\left[\frac{B_N\cdot {\left(T_L\right)}^2}{N}\right]\backslash mu\_\{n\}^\{I\}\; =\backslash frac\{\{A\_N\}\backslash cdot\backslash left(T\_\{L\}\backslash right)^\{3\; /\; 2\}\}\{N\}\; f\backslash left[\backslash frac\{\{B\_N\}\backslash cdot\backslash left(T\_\{L\}\backslash right)^\{2\}\}\{N\}\backslash right]$$ $${\mu }_p^I=\frac{A_P\cdot {\left(T_L\right)}^{3\mathrm{/}2}}{N}f\left[\frac{B_P\cdot {\left(T_L\right)}^2}{N}\right]\backslash mu\_\{p\}^\{I\}\; =\backslash frac\{\{A\_P\}\backslash cdot\backslash left(T\_\{L\}\backslash right)^\{3\; /\; 2\}\}\{N\}\; f\backslash left[\backslash frac\{\{B\_P\}\backslash cdot\backslash left(T\_\{L\}\backslash right)^\{2\}\}\{N\}\backslash right]$$

Here, $NN$ is the total concentration, $T_{L}T\_\{L\}$ is the lattice temperature and $nn$,$pp$ are the electron and hole carrier concentrations, respectively .

$$f(x)={[\ln (1+x)-\frac{x}{1+x}]}^{-1}f(x)=\backslash left[\backslash ln\; (1+x)-\backslash frac\{x\}\{1+x\}\backslash right]^\{-1\}$$

The values of the lattice scattering terms, ${\mu }_{n,p}^L\backslash mu\_\{\{n,p\}\}^\{L\}$ are defined by following equations.

$${\mu }_{n,p}^L={\mu }_{L0}{\left(\frac{T_L}{300}\right)}^{-\alpha }\backslash mu\_\{\; \{n,\; p\}\}^\{L\}=\backslash mu\_\{L0\}\backslash left(\backslash frac\{T\_\{L\}\}\{300\}\backslash right)^\{-\backslash alpha\}$$

The parameters for this model with their default values (Phosphorus-doped Silicon for electron values and Boron-doped Silicon for hole values) are defined in the following table:

Symbol	Parameter Name	Electron Value	Hole Value	Units
μL0	muL0	0.1430	0.0495	m2/(V*s)
α	alpha	2.2	2.2	N/A
A	A	4.61×1023	1.0×1023	m-3
B	B	1.52×1021	6.25×1020	m-3

The type of corresponding parameters are listed in the following table:

Symbol	Updated During Simulation	Predefinition of Material Parameter	Device Configuration	Description
μL0	✗	✓	✗	N/A
α	✗	✓	✗	N/A
A	✗	✓	✗	N/A
B	✗	✓	✗	N/A
TL	✗	✗	✓	Lattice temperature (K)
N	✗	✗	✓	Total doping concentration (m-3)
n	✗	✗	✓	Electron concentration (m-3)
p	✗	✗	✓	Hole concentration (m-3)

Reference:

[1] Dorkel, J. M., and Ph Leturcq. "Carrier mobilities in silicon semi-empirically related to temperature, doping and injection level." Solid-State Electronics 24.9 (1981): 821-825. doi: 10.1016/0038-1101(81)90097-6

1.7 Klaassen Model

Klaassen's model is used to calculate ${\mu }_0\backslash mu\_\{0\}$.

The model by D. B. M. Klaassen^[1,2] includes the effects of lattice scattering, impurity scattering, carrier-carrier scattering, and impurity clustering effects at high concentration.

The total mobility can be described by its components using Matthiessen’s rule as:

$${\mu }_0^{-1}={\mu }_L^{-1}+{\mu }_{DAP}{}^{-1}\backslash mu\_\{0\}^\{-1\}=\backslash mu\_\{L\}^\{-1\}+\backslash mu\_\{D\; A\; P\}\{\; \}^\{-1\}$$

where ${\mu }_L\backslash mu\_\{L\}$ are the electron and hole mobilities due to lattice scattering, ${\mu }_{\text{DAP}}\backslash mu\_\{\backslash text\{DAP\}\}$ are the electron and hole mobilities due to donor (D), acceptor (A), screening (P) and carrier-carrier scattering.

$${\mu }_L={\mu }_{max}\ast (\frac{300}{T_L}{)}^{\theta }\backslash mu\_L\; =\; \backslash mu\_\{max\}*(\backslash frac\{300\}\{T\_L\})^\backslash theta$$

The impurity-carrier scattering components of the total mobility are given by:

$${\mu }_{DAP}={\mu }_N\frac{N_{sc}}{N_{sc,\text{ eff }}}{\left(\frac{N_{ref1}}{N_{sc}}\right)}^{\alpha }+{\mu }_c\left(\frac{n+p}{N_{sc,\text{ eff }}}\right)\backslash mu\_\{D\; A\; P\}=\backslash mu\_\{N\}\; \backslash frac\{N\_\{s\; c\}\}\{N\_\{\; s\; c,\; \backslash text\; \{\; eff\; \}\}\}\backslash left(\backslash frac\{N\_\{ref1\}\}\{N\_\{s\; c\}\}\backslash right)^\{\backslash alpha\}+\backslash mu\_\{c\}\backslash left(\backslash frac\{n+p\}\{N\_\{\; s\; c,\; \backslash text\; \{\; eff\; \}\}\}\backslash right)$$

The impurity scattering component, ${\mu }_N\backslash mu\_\{N\}$ ,is given by:

$${\mu }_N=\frac{{\mu }_{max}^2}{{\mu }_{max}-{\mu }_{min}}\cdot (\frac{T_L}{300}{)}^{3\alpha -1.5}\backslash mu\_N\; =\; \backslash frac\{\backslash mu\_\{max\}^\{2\}\}\{\backslash mu\_\{max\}-\backslash mu\_\{min\}\}\backslash cdot(\backslash frac\{T\_L\}\{300\})^\{3\backslash alpha-1.5\}$$

The carrier-carrier scattering component, ${\mu }_c\backslash mu\_c$, is given by:

$${\mu }_c=\frac{{\mu }_{min}\ast {\mu }_{max}}{{\mu }_{max}-{\mu }_{min}}\cdot (\frac{300}{T_L}{)}^{0.5}\backslash mu\_c\; =\; \backslash frac\{\backslash mu\_\{min\}*\backslash mu\_\{max\}\}\{\backslash mu\_\{max\}-\backslash mu\_\{min\}\}\backslash cdot(\backslash frac\{300\}\{T\_L\})^\{0.5\}$$

The $N_{sc}N\_\{sc\}$ parameter is given by:

$$N_{sc}=N_D+N_A+pforelectronN\_\{sc\}\; =\; N\_D+N\_A+p\; \backslash qquad\; for\; \backslash \; \backslash \; electron$$ $$N_{sc}=N_D+N_A+nforholeN\_\{sc\}\; =\; N\_D+N\_A+n\; \backslash qquad\; for\; \backslash \; \backslash \; hole$$

where $N_{D}N\_\{D\}$ is the donor concentration in $m^{-3}m^\{-3\}$, $N_{A}N\_\{A\}$ is the acceptor concentration in $m^{-3}m^\{-3\}$, $nn$ is the electron concentration in $m^{-3}m^\{-3\}$ and $pp$ is the hole concentration in $m^{-3}m^\{-3\}$.

The $N_{sc,eff}N\_\{\{sc,\; eff\}\}$ parameter is given by:

$$N_{sc,eff}=N_D+G(P_n)\cdot N_A+(\frac{p}{F(P_n)})forelectronsN\_\{sc,\; eff\}\; =\; N\_D+G(P\_n)\backslash cdot\; N\_A+(\backslash frac\{p\}\{F(P\_n)\})\; \backslash quad\; for\backslash \; \backslash \; electrons$$ $$N_{sc,eff}=N_A+G(P_p)\cdot N_D+(\frac{n}{F(P_p)})forholesN\_\{sc,\; eff\}\; =\; N\_A+G(P\_p)\backslash cdot\; N\_D+(\backslash frac\{n\}\{F(P\_p)\})\; \backslash quad\; for\backslash \; \backslash \; holes$$

The function, $G(P_n)G(P\_\{n\})$ and $G(P_p)G(P\_\{p\})$ are given by:

$$G(P_n)=1-\frac{s_1}{[s_2+P_n\cdot (\frac{(T_L\mathrm{/}300)}{m_e}{)}^{s_4}{]}^{s_3}}+\frac{s_5}{[P_n\cdot (\frac{m_e}{(T_L\mathrm{/}300)}{)}^{s_7}{]}^{s_6}}G(P\_n)\; =\; 1-\backslash frac\{s\_1\}\{[s\_2+P\_n\backslash cdot\; (\backslash frac\{(T\_L/300)\}\{m\_e\})^\{s\_4\}]^\{s\_3\}\}+\backslash frac\{s\_5\}\{[P\_n\; \backslash cdot\; (\backslash frac\{m\_e\}\{(T\_L/300)\})^\{s\_7\}]^\{s\_6\}\}$$ $$G(P_p)=1-\frac{s_1}{[s_2+P_p\cdot (\frac{(T_L\mathrm{/}300)}{m_h}{)}^{s_4}{]}^{s_3}}+\frac{s_5}{[P_p\cdot (\frac{m_h}{(T_L\mathrm{/}300)}{)}^{s_7}{]}^{s_6}}G(P\_p)\; =\; 1-\backslash frac\{s\_1\}\{[s\_2+P\_p\backslash cdot\; (\backslash frac\{(T\_L/300)\}\{m\_h\})^\{s\_4\}]^\{s\_3\}\}+\backslash frac\{s\_5\}\{[P\_p\; \backslash cdot\; (\backslash frac\{m\_h\}\{(T\_L/300)\})^\{s\_7\}]^\{s\_6\}\}$$

Here, $m_{e}m\_\{e\}$ and $m_{h}m\_\{h\}$ are the ratio of electron and hole effective masses to free masses and the parameters $s_{1}s\_\{1\}$ through $s_{7}s\_\{7\}$ are user-specifiable model parameters.

The function, $F(P_n)F(P\_n)$ and $F(P_p)F(P\_p)$ are given by:

$$F(P_n)=\frac{r_1\cdot P_n^{r_6}+r_2+r_3\cdot \frac{m_e}{m_h}}{P_n^{r_6}+r_4+r_5\cdot \frac{m_e}{m_h}}F(P\_n)\; =\; \backslash frac\{r\_1\; \backslash cdot\; P\_n^\{r\_6\}\; +\; r\_2\; +\; r\_3\; \backslash cdot\; \backslash frac\{m\_e\}\{m\_h\}\}\{P\_n^\{r\_6\}+r\_4+r\_5\backslash cdot\; \backslash frac\{m\_e\}\{m\_h\}\}$$ $$F(P_p)=\frac{r_1\cdot P_p^{r_6}+r_2+r_3\cdot \frac{m_h}{m_e}}{P_p^{r_6}+r_4+r_5\cdot \frac{m_h}{m_e}}F(P\_p)\; =\; \backslash frac\{r\_1\; \backslash cdot\; P\_p^\{r\_6\}\; +\; r\_2\; +\; r\_3\; \backslash cdot\; \backslash frac\{m\_h\}\{m\_e\}\}\{P\_p^\{r\_6\}+r\_4+r\_5\backslash cdot\; \backslash frac\{m\_h\}\{m\_e\}\}$$

where the parameters, $r_{1}r\_\{1\}$ through $r_{6}r\_\{6\}$, are user-specifiable model parameters.

The screening parameters, $P_{n}P\_\{n\}$ and $P_{p}P\_\{p\}$, are given by:

$$P_n=[\frac{F_{CW}}{P_{CW,n}}+\frac{F_{BH}}{P_{BH,n}}{]}^{-1}P\_n\; =\; [\backslash frac\{F\_\{CW\}\}\{P\_\{CW,\; n\}\}+\backslash frac\{F\_\{BH\}\}\{P\_\{BH,\; n\}\}]^\{-1\}$$ $$P_p=[\frac{F_{CW}}{P_{CW,p}}+\frac{F_{BH}}{P_{BH,p}}{]}^{-1}P\_p\; =\; [\backslash frac\{F\_\{CW\}\}\{P\_\{CW,\; p\}\}+\backslash frac\{F\_\{BH\}\}\{P\_\{BH,\; p\}\}]^\{-1\}$$

Here, the $F_{CW}F\_\{CW\}$ and $F_{BH}F\_\{BH\}$ parameters are user-specifiable model parameters.

The functions, $P_{BH,n}P\_\{BH,n\}$ and $P_{BH,p}P\_\{BH,p\}$, $P_{CW,n}P\_\{CW,n\}$, and $P_{CW,p}P\_\{CW,p\}$ are given by the following equations.

$$P_{BH,n}=\frac{1.36\ast 10^{20}}{n+p}\cdot m_e\cdot (\frac{T_L}{300}{)}^{2}P\_\{BH,n\}\; =\; \backslash frac\{1.36*10^\{20\}\}\{n+p\}\; \backslash cdot\; m\_e\; \backslash cdot\; (\backslash frac\{T\_L\}\{300\})^2$$ $$P_{BH,p}=\frac{1.36\ast 10^{20}}{n+p}\cdot m_h\cdot (\frac{T_L}{300}{)}^{2}P\_\{BH,p\}\; =\; \backslash frac\{1.36*10^\{20\}\}\{n+p\}\; \backslash cdot\; m\_h\; \backslash cdot\; (\backslash frac\{T\_L\}\{300\})^2$$ $$P_{CW,n}=3.97\ast 10^{13}\cdot [\frac{1}{Z_n^3\cdot N_{sc}}\cdot (\frac{T_L}{300}{)}^3{]}^{\frac{2}{3}}P\_\{CW,n\}\; =\; 3.97*10^\{13\}\; \backslash cdot\; [\backslash frac\{1\}\{Z\_n^3\; \backslash cdot\; N\_\{sc\}\}\; \backslash cdot\; (\backslash frac\{T\_L\}\{300\})^3]^\{\backslash frac\{2\}\{3\}\}$$ $$P_{CW,p}=3.97\ast 10^{13}\cdot [\frac{1}{Z_p^3\cdot N_{sc}}\cdot (\frac{T_L}{300}{)}^3{]}^{\frac{2}{3}}P\_\{CW,p\}\; =\; 3.97*10^\{13\}\; \backslash cdot\; [\backslash frac\{1\}\{Z\_p^3\; \backslash cdot\; N\_\{sc\}\}\; \backslash cdot\; (\backslash frac\{T\_L\}\{300\})^3]^\{\backslash frac\{2\}\{3\}\}$$

$Z_{n}Z\_n$ and $Z_{p}Z\_p$ are clustering functions given by:

$$Z_n=1+\frac{1}{C_D+(\frac{N_{ref,D}}{N_D}{)}^2}Z\_n\; =\; 1\; +\; \backslash frac\{1\}\{C\_D\; +\; (\backslash frac\{N\_\{ref,\; D\}\}\{N\_D\})^2\}$$ $$Z_p=1+\frac{1}{C_A+(\frac{N_{ref,A}}{N_A}{)}^2}Z\_p\; =\; 1\; +\; \backslash frac\{1\}\{C\_A\; +\; (\backslash frac\{N\_\{ref,\; A\}\}\{N\_A\})^2\}$$

where $C_{D}C\_D$, $C_{A}C\_A$, $N_{ref,D}N\_\{ref,\; D\}$, and $N_{ref,A}N\_\{ref,A\}$ are user-definable parameters.

The parameters for this model with their default values (Arsenic-doped Silicon for electron values and Boron-doped Silicon for hole values) are defined in the following table:

Symbol	Parameter Name	Electron Value	Hole Value	Units
μmax	mumax	0.1417	0.04705	m2/(V*s)
μmin	mumin	0.00522	0.00449	m2/(V*s)
θ	theta	2.285	2.247	N/A
α	alpha	0.68	0.719	N/A
Nref1	Nref1	9.68×1022	2.23×1023	m-3
me	me	1.0	1.0	N/A
mh	mh	1.258	1.258	N/A
s1	s1	0.89233	0.89233	N/A
s2	s2	0.41372	0.41372	N/A
s3	s3	0.19778	0.19778	N/A
s4	s4	0.28227	0.28227	N/A
s5	s5	0.005978	0.005978	N/A
s6	s6	1.80618	1.80618	N/A
s7	s7	0.72169	0.72169	N/A
r1	r1	0.7643	0.7643	N/A
r2	r2	2.2999	2.2999	N/A
r3	r3	6.5502	6.5502	N/A
r4	r4	2.3670	2.3670	N/A
r5	r5	-0.8552	-0.8552	N/A
r6	r6	0.6478	0.6478	N/A
FCW	FCW	2.459	2.459	N/A
FBH	FBH	3.828	3.828	N/A
C	C	0.21	0.50	N/A
Nref	Nref	4.0×1026	7.2×1026	m-3

The type of corresponding parameters are listed in the following table:

Symbol	Updated During Simulation	Predefinition of Material Parameter	Device Configuration	Description
μmax	✗	✓	✗	N/A
μmin	✗	✓	✗	N/A
θ	✗	✓	✗	N/A
α	✗	✓	✗	N/A
Nref1	✗	✓	✗	N/A
s1	✗	✓	✗	N/A
s2	✗	✓	✗	N/A
s3	✗	✓	✗	N/A
s4	✗	✓	✗	N/A
s5	✗	✓	✗	N/A
s6	✗	✓	✗	N/A
s7	✗	✓	✗	N/A
r1	✗	✓	✗	N/A
r2	✗	✓	✗	N/A
r3	✗	✓	✗	N/A
r4	✗	✓	✗	N/A
r5	✗	✓	✗	N/A
r6	✗	✓	✗	N/A
FCW	✗	✓	✗	N/A
FBH	✗	✓	✗	N/A
C	✗	✓	✗	N/A
Nref	✗	✓	✗	N/A
NA	✗	✗	✓	Acceptor concentration (m-3)
ND	✗	✗	✓	Donor concentration (m-3)
TL	✗	✗	✓	Lattice temperature (K)
n	✗	✗	✓	Electron concentration (m-3)
p	✗	✗	✓	Hole concentration (m-3)
me	✗	✗	✓	The ratio of electron effective mass to free mass
mh	✗	✗	✓	The ratio of hole effective mass to free mass

Issues:

1. Minority electron mobility is not fitting well (Fig. 6 in reference 1)

2. The value of parameter r5 in reference 1 is -0.01552. In C software and S software, r5 is -0.8552.

Reference:

[1] Klaassen, D. B. M. "A unified mobility model for device simulation—I. Model equations and concentration dependence." Solid-State Electronics 35.7 (1992): 953-959. doi: 10.1016/0038-1101(92)90325-7

[2] Klaassen, D. B. M. "A unified mobility model for device simulation—II. Temperature dependence of carrier mobility and lifetime." Solid-State Electronics 35.7 (1992): 961-967. doi: 10.1016/0038-1101(92)90326-8

1.8 Lombardi CVT Model

This model is used to calculate ${\mu }_0\backslash mu\_0$.

Lombardi CVT model^[1] includes the transverse field, doping dependent and temperature dependent parts of the mobility. These components , ${\mu }_{AC}\backslash mu\_\{AC\}$, ${\mu }_{sr}\backslash mu\_\{sr\}$ and ${\mu }_b\backslash mu\_\{b\}$, are combined using Matthiessen’s rule as follows:

$${\mu }_0^{-1}={\mu }_{AC}^{-1}+{\mu }_b^{-1}+{\mu }_{sr}^{-1}\backslash mu\_0^\{-1\}\; =\; \backslash mu\_\{AC\}^\{-1\}\; +\; \backslash mu\_\{b\}^\{-1\}\; +\; \backslash mu\_\{sr\}^\{-1\}$$

The first term, ${\mu }_{AC}\backslash mu\_\{AC\}$, is the effect of surface phonon scattering:

$${\mu }_{AC}=\frac{B}{(\frac{E_{\perp }}{E_1}{)}^E}+\left(\frac{C\cdot (\frac{N}{N_1}{)}^{\tau }}{(\frac{E_{\perp }}{E_1}{)}^D}\right)\cdot \frac{T_1}{T_L}\backslash mu\_\{AC\}\; =\; \backslash frac\{B\}\{(\backslash frac\{E\_\{\backslash perp\}\}\{E\_1\})^E\}+\backslash left(\backslash frac\{C\; \backslash cdot\; (\; \backslash frac\{N\}\{N\_1\})^\{\backslash tau\}\}\{(\backslash frac\{E\_\{\backslash perp\}\}\{E\_1\})^D\}\; \backslash right)\; \backslash cdot\; \backslash frac\{T\_1\}\{T\_L\}$$

where $T_{L}T\_L$ is the lattice temperature, $E_{\perp }E\_\{\backslash perp\}$ is the electric field components perpendicular to the current density vector, and $NN$ is the total doping concentration. In $E_{1}E\_1$ is $1V\mathrm{/}m1V/m$, $N_{1}N\_1$ is $1m^{-3}1m^\{-3\}$, and $T_{1}T\_1$ is $1K1K$. The parameters, $BB$, $CC$, $DD$, $EE$, $\tau \backslash tau$, are user-defined.

The second term, ${\mu }_{sr}\backslash mu\_\{sr\}$, is the effect of surface roughness and is given by:

$$\frac{1}{{\mu }_{sr}}=\frac{(\frac{E_{\perp }}{E_1}{)}^K}{DEL}+\frac{(\frac{E_{\perp }}{E_1}{)}^3}{FEL}\backslash frac\{1\}\{\backslash mu\_\{sr\}\}\; =\; \backslash frac\{(\backslash frac\{E\_\{\backslash perp\}\}\{E\_1\})^K\}\{DEL\}\; +\; \backslash frac\{(\backslash frac\{E\_\{\backslash perp\}\}\{E\_1\})^3\}\{FEL\}$$

The default values of $FELFEL$ are set so high that the second term can be ignored. The $KK$, $DELDEL$, $FELFEL$ parameters are user-definable.

The third mobility component, ${\mu }_b\backslash mu\_b$, is the effect of scattering with optical intervalley phonons and is given by:

$${\mu }_b={\mu }_0\cdot exp(\frac{-P_C}{N})+\frac{[{\mu }_{max}\cdot (\frac{T_L}{300}{)}^{-\gamma }-{\mu }_0]}{1+(\frac{N}{C_R}{)}^{\alpha }}-\frac{{\mu }_1}{1+(\frac{C_S}{N}{)}^{\beta }}\backslash mu\_b\; =\; \backslash mu\_0\; \backslash cdot\; exp(\backslash frac\{-P\_C\}\{N\})\; +\; \backslash frac\{[\backslash mu\_\{max\}\; \backslash cdot\; (\backslash frac\{T\_L\}\{300\})^\{-\backslash gamma\}-\backslash mu\_0]\}\{1+(\backslash frac\{N\}\{C\_R\})^\{\backslash alpha\}\}-\backslash frac\{\backslash mu\_\{1\}\}\{1+(\backslash frac\{C\_S\}\{N\})^\{\backslash beta\}\}$$

Here, $NN$ is the total density of impurities.

Due to the fact that ${\mu }_{AC}\backslash mu\_\{AC\}$ and ${\mu }_{sr}\backslash mu\_\{sr\}$ are related to interaction with an interface, you can specify the $L_{CRIT}L\_\{CRIT\}$ for electrons or holes to modify this interaction. The total mobility can be modified to

$$\frac{1}{{\mu }_0}=exp(-\frac{l}{L_{CRIT}})\cdot (\frac{1}{{\mu }_{AC}}+\frac{1}{{\mu }_{sr}})+\frac{1}{{\mu }_b}\backslash frac\{1\}\{\backslash mu\_0\}\; =\; exp(-\backslash frac\{l\}\{L\_\{CRIT\}\})\; \backslash cdot(\backslash frac\{1\}\{\backslash mu\_\{AC\}\}\; +\; \backslash frac\{1\}\{\backslash mu\_\{sr\}\})\; +\; \backslash frac\{1\}\{\backslash mu\_b\}$$

where $ll$ is the distance to the nearest interface from the point where the mobility will be calculated.

The parameters for this model with their default values (n-MOSFET for electron values and p-MOSFET for hole values) are defined in the following table:

Symbol	Parameter Name	Electron Value	Hole Value	Units
α	alpha	0.68	0.71	N/A
β	beta	2.0	2.0	N/A
B	B	4.75×103	9.925×102	m2/(V*s)
C	C	17.4	88.42	m2/(V*s)
CR	CR	9.68×1022	2.23×1023	m-3
CS	CS	3.43×1026	6.10×1026	m-3
DEL	DEL	5.82×1010	2.055×1010	m2/(V*s)
D	D	0.333	0.333	N/A
E	E	1.0	1.0	N/A
FEL	FEL	1.0×1054	1.0×1054	m2/(V*s)
K	K	2.0	2.0	N/A
γ	gamma	2.5	2.2	N/A
μ0	mu0	0.00522	0.00449	m2/(V*s)
μ1	mu1	0.00434	0.00290	m2/(V*s)
μmax	mumax	0.1417	0.04705	m2/(V*s)
Lcrit	Lcrit	0.01	0.01	m
PC	Pc	0.0	9.23×1022	m-3
τ	tau	0.125	0.0317	N/A

The type of corresponding parameters are listed in the following table:

Symbol	Updated During Simulation	Predefinition of Material Parameter	Device Configuration	Description
E⊥	✓	✗	✗	Transverse electric field (V/m)
l	✓	✗	✗	The distance to the nearest interface from the simulation point (m)
α	✗	✓	✗	N/A
β	✗	✓	✗	N/A
B	✗	✓	✗	N/A
C	✗	✓	✗	N/A
CR	✗	✓	✗	N/A
CS	✗	✓	✗	N/A
DEL	✗	✓	✗	N/A
D	✗	✓	✗	N/A
E	✗	✓	✗	N/A
FEL	✗	✓	✗	N/A
K	✗	✓	✗	N/A
γ	✗	✓	✗	N/A
μ0	✗	✓	✗	N/A
μ1	✗	✓	✗	N/A
μmax	✗	✓	✗	N/A
PC	✗	✓	✗	N/A
τ	✗	✓	✗	N/A
N	✗	✗	✓	Total concentration (m-3)
Lcrit	✗	✗	✓	N/A
TL	✗	✗	✓	Lattice temperature (K)

Reference:

[1] Lombardi, Claudio, et al. "A physically based mobility model for numerical simulation of nonplanar devices." IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 7.11 (1988): 1164-1171. doi: [10.1109/43.9186]

1.9 Masetti Model

Masetti model is used to calculate ${\mu }_0\backslash mu\_0$.

This model implements the following relationship^[1]:

$${\mu }_0={\mu }_{\min 1}{e}^{-\frac{P_c}{N_i}}+\frac{{\mu }_{\max }{\left(\frac{T_L}{T_0}\right)}^{-\zeta }-{\mu }_{\min 2}}{1+{\left(\frac{N_i}{C_r}\right)}^{\alpha }}-\frac{{\mu }_1}{1+{\left(\frac{C_s}{N_i}\right)}^{\beta }}\backslash mu\_\{0\}=\backslash mu\_\{\backslash min\; 1\}\; e^\{-\backslash frac\{P\_\{c\}\}\{N\_\{i\}\}\}+\backslash frac\{\backslash mu\_\{\backslash max\; \}\backslash left(\backslash frac\{T\_L\}\{T\_\{0\}\}\backslash right)^\{-\backslash zeta\}-\backslash mu\_\{\backslash min\; 2\}\}\{1+\backslash left(\backslash frac\{N\_\{i\}\}\{C\_\{r\}\}\backslash right)^\{\backslash alpha\}\}-\backslash frac\{\backslash mu\_\{1\}\}\{1+\backslash left(\backslash frac\{C\_\{s\}\}\{N\_\{i\}\}\backslash right)^\{\backslash beta\}\}$$

where $N_{i}N\_i$ is the total local dopant concentration.

Symbol	Parameter Name	Electron Value	Hole Value	Units
μmax	mumax	0.1417	0.04705	m2/(V*s)
μmin1	mumin1	0.00522	0.00449	m2/(V*s)
μmin2	mumin2	0.00522	0	m2/(V*s)
μ1	mu1	0.00434	0.0029	m2/(V*s)
ζ	zeta	2.5	2.2	N/A
Pc	Pc	0	9.3×1022	m-3
Cr	Cr	9.68×1022	2.23×1023	m-3
Cs	Cs	3.34×1026	6.1×1026	m-3
α	alpha	0.68	0.719	N/A
β	beta	2.0	2.0	N/A

The type of corresponding parameters are listed in the following table:

Symbol	Updated During Simulation	Predefinition of Material Parameter	Device Configuration	Description
μmax	✗	✓	✗	N/A
μmin1	✗	✓	✗	N/A
μmin2	✗	✓	✗	N/A
μ1	✗	✓	✗	N/A
ζ	✗	✓	✗	N/A
Pc	✗	✓	✗	N/A
Cr	✗	✓	✗	N/A
Cs	✗	✓	✗	N/A
α	✗	✓	✗	N/A
β	✗	✓	✗	N/A
Ni	✗	✗	✓	Total concentration (m-3)
TL	✗	✗	✓	Lattice temperature (K)

Reference:

[1] Masetti, Guido, Maurizio Severi, and Sandro Solmi. "Modeling of carrier mobility against carrier concentration in arsenic-, phosphorus-, and boron-doped silicon." IEEE Transactions on electron devices 30.7 (1983): 764-769. doi: 10.1109/T-ED.1983.21207

1.10 Shirahata’s Mobility Model

The Shirahata Mobility Model is used to calculate ${\mu }_0\backslash mu\_0$.

The Shirahata Mobility Model^[1] is a general purpose MOS mobility model that accounts for screening effects in the inversion layer.

$$\mu =\frac{{\mu }_0\cdot (\frac{T_L}{300}{)}^{-\theta }}{[1+\frac{\mathrm{\mid }{E}_{\perp }\mathrm{\mid }}{E_1}{]}^{P_1}+[\frac{\mathrm{\mid }{E}_{\perp }\mathrm{\mid }}{E_2}{]}^{P_2}}\backslash mu\; =\; \backslash frac\{\backslash mu\_0\; \backslash cdot\; (\backslash frac\{T\_L\}\{300\})^\{-\backslash theta\}\}\{[1+\backslash frac\{|E\_\{\backslash perp\}|\}\{E\_1\}]^\{P\_1\}+[\backslash frac\{|E\_\{\backslash perp\}|\}\{E\_2\}]^\{P\_2\}\}$$

where $E_{\perp }E\_\{\backslash perp\}$ is the electric field components perpendicular to the current density vector.

The parameters for this model with their default values (n-MOSFET for electron and p-MOSFET for hole) are defined in the following table:

Symbol	Parameter Name	Electron Value	Hole Value	Units
μ0	mu0	0.1430	0.0500	m2/(V*s)
E1	E1	6.3×105	8.0×105	V/m
E2	E2	7.7×107	3.9×107	V/m
P1	P1	0.28	0.3	N/A
P2	P2	2.9	1.0	N/A
θ	theta	2.285	2.247	N/A

The type of corresponding parameters are listed in the following table:

Symbol	Updated During Simulation	Predefinition of Material Parameter	Device Configuration	Description
E⊥	✓	✗	✗	Transverse electric field (V/m)
μ0	✗	✓	✗	N/A
E1	✗	✓	✗	N/A
E2	✗	✓	✗	N/A
P1	✗	✓	✗	N/A
P2	✗	✓	✗	N/A
θ	✗	✓	✗	N/A
TL	✗	✗	✓	Lattice temperature (K)

Reference:

[1] Shirahata, Masayoshi, et al. "A mobility model including the screening effect in MOS inversion layer." IEEE transactions on computer-aided design of integrated circuits and systems 11.9 (1992): 1114-1119. doi: 10.1109/43.159997

1.11 Watt Model

This model is used to calculate ${\mu }_0\backslash mu\_0$.

Watt Model^[1] includes phonon scattering, surface roughness scattering and charged impurity scattering mechanisms.

The phonon and surface roughness components are functions of effective electric field. The charged impurity component is a function of the channel doping density.

The effective mobilities:

$$\frac{1}{{\mu }_0}=\frac{1}{M_{ref1}}(\frac{10^6}{E_{eff}}{)}^{{\alpha }_1}+\frac{1}{M_{ref2}}(\frac{10^6}{E_{eff}}{)}^{{\alpha }_2}+\frac{1}{M_{ref3}}(\frac{10^{18}}{N_B}{)}^{-1}(\frac{10^{12}}{N_i}{)}^{{\alpha }_3}\backslash frac\{1\}\{\backslash mu\_\{0\}\}\; =\; \backslash frac\{1\}\{M\_\{ref1\}\}(\backslash frac\{10^6\}\{E\_\{eff\}\})^\{\backslash alpha\_1\}\; +\; \backslash frac\{1\}\{M\_\{ref2\}\}(\backslash frac\{10^6\}\{E\_\{eff\}\})^\{\backslash alpha\_2\}\; +\; \backslash frac\{1\}\{M\_\{ref3\}\}(\backslash frac\{10^\{18\}\}\{N\_B\})^\{-1\}(\backslash frac\{10^\{12\}\}\{N\_i\})^\{\backslash alpha\_3\}$$

Here, $N_{B}N\_B$ is the surface trapped charge density, $N_{i}N\_i$ is the inversion layer charge density and $E_{eff}E\_\{eff\}$ is the effective electric field given by:

$$E_{eff}=E_{\perp }+E_{TA}\cdot (E_0-E_{\perp })E\_\{eff\}\; =\; E\_\{\backslash perp\}\; +\; E\_\{TA\}\; \backslash cdot(E\_0-E\_\{\backslash perp\})$$

where $E_{\perp }E\_\{\backslash perp\}$ is the electric field perpendicular to the current flow and $E_{0}E\_0$ is the perpendicular electric field at the insulator-semiconductor interface. The parameter $E_{TA}E\_\{TA\}$ is user-definable.

The parameters for this model with their default values (n-MOSFET for electron values and p-MOSFET for hole values) are defined in the following table:

Symbol	Parameter Name	Electron Value	Hole Value	Units
ETA	ETA	0.50	0.33	N/A
Mref1	Mref1	0.0481	0.00928	m2/(V*s)
Mref2	Mref2	0.0591	0.0124	m2/(V*s)
Mref3	Mref3	0.1270	0.0534	m2/(V*s)
α1	alpha1	-0.16	-0.296	N/A
α2	alpha2	-2.17	-1.62	N/A
α3	alpha3	1.07	1.02	N/A

The type of corresponding parameters are listed in the following table:

Symbol	Updated During Simulation	Predefinition of Material Parameter	Device Configuration	Description
E⊥	✓	✗	✗	Transverse electric field (V/m)
E0	✓	✗	✗	Transverse electric field at the interface (V/m)
ETA	✗	✓	✗	N/A
Mref1	✗	✓	✗	N/A
Mref2	✗	✓	✗	N/A
Mref3	✗	✓	✗	N/A
α1	✗	✓	✗	N/A
α2	✗	✓	✗	N/A
α3	✗	✓	✗	N/A
Ni	✗	✗	✓	Total concentration (m-3)
NB	✗	✗	✓	Surface trapped charge density (m-3)

1.11.1 Modified Watt Model

Parameters, $Y_{max}Y\_\{max\}$, is the maximum value of the $YY$ coordinate.

Parameters, $X_{min}X\_\{min\}$ and $X_{max}X\_\{max\}$, is the range of the model in the X direction.

The effective normal electric field can be modified as following:

$$E_{\perp }=E_y\cdot exp\left(\frac{-(y-y_{int})}{Y_{CHAR}}\right)E\_\{\backslash perp\}\; =\; E\_y\; \backslash cdot\; exp\backslash left(\backslash frac\{-(y-y\_\{int\})\}\{Y\_\{CHAR\}\}\backslash right)$$

where $E_{\perp }E\_\{\backslash perp\}$ is the transverse electric field, $E_{y}E\_y$ is the transverse electric field at the interface, $yy$ is the local $YY$ coordinate and $y_{int}y\_\{int\}$ is the $YY$ coordinate of the interface. The $Y_{CHAR}Y\_\{CHAR\}$ parameters are user-definable.

The parameters for this model with their default values are defined in the following table:

Symbol	Parameter Name	Electron Value	Hole Value	Units
Xmin	Xmin	-1.0×1030	-1.0×1030	m
Xmax	Xmax	1.0×1030	1.0×1030	m
Ymax	Ymax	-1.0×1030	-1.0×1030	m
YCHAR	YCHAR	1.0×1030	1.0×1030	m

The type of corresponding parameters are listed in the following table:

Symbol	Updated During Simulation	Predefinition of Material Parameter	Device Configuration	Description
Ey	✓	✗	✗	Transverse electric field at the interface (V/m)
y	✓	✗	✗	The local Y coordinate
Xmin	✗	✗	✓	Minimum value of the X coordinate (m)
Xmax	✗	✗	✓	Maximum value of the X coordinate (m)
Ymax	✗	✗	✓	Maximum value of the Y coordinate (m)
YCHAR	✗	✗	✓	N/A
yint	✗	✗	✓	The Y coordinate of the interface (m)

Issues:

1. The default parameters fit very low doping concentration well, but those parameters still need to fit with a higher doping concentration( Fig. 4 in ref1 )

Reference:

[1] Watt, Jeffrey Thomas. Modeling the performance of liquid-nitrogen-cooled CMOS VLSI. Stanford University, 1989.

Here marks the end of the low field mobility models discussion.
Please proceed to the next article for high field mobility models of electrons and holes.

2. Electron/Hole High-Field Mobility Model

High-field mobility models come into play when the electric field strength is significant, causing the charge carriers to experience velocity saturation. In such scenarios, the mobility of electrons and holes deviates from the linear relationship, requiring more complex models to accurately describe their behavior.

2.1 Constant Mobilities

The simplest mobility model uses constant mobilities ${\mu }_{0n}\backslash mu\_\{0n\}$ and ${\mu }_{0p}\backslash mu\_\{0p\}$ for electrons and holes, respectively, throughout each material region in the device.

The parameters for this model with their default values are defined in the following table:

Symbol	Parameter Name	Electron Value	Hole Value	Units
μ0	mu0	0.1	0.05	m2/(V*s)

2.2 Two-piece mobility model

This model is used to calculate $\mu \backslash mu$.

Two-piece mobility model is a field dependent mobility model and is given by:

$$\mu =v_{sat}\mathrm{/}FforF\ge F_0\backslash mu\; =\; v\_\{sat\}\; /\; F\; \backslash quad\; for\; \backslash \; \backslash \; F\; \backslash geq\; F\_\{0\}$$ $$v_{sat}={\mu }_0{F}_{0}v\_\{sat\}\; =\; \backslash mu\_\{0\}\; F\_\{0\}$$

where $F_{0}F\_\{0\}$ is a threshold field beyond which the carrier velocity saturates to a constant and $v_{sat}v\_\{sat\}$ is saturation velocity.

The type of corresponding parameters are listed in the following table:

Symbol	Updated During Simulation	Predefinition of Material Parameter	Device Configuration	Description
F	✓	✗	✗	Parallel electric field (V/m)
μ0	✗	✓	✗	Low-field mobility (m2/(V*s))
vsat	✗	✓	✗	Saturation velocity (m/s)

2.3 Canali Model

This model is used to calculate $\mu \backslash mu$.

The Canali or beta model^[1,2] is given by:

$$\mu =\frac{{\mu }_0}{{(1+{\left({\mu }_{0}F\mathrm{/}{v}_s\right)}^{\beta })}^{1\mathrm{/}\beta }}\backslash mu=\backslash frac\{\backslash mu\_\{0\; \}\}\{\backslash left(1+\backslash left(\backslash mu\_\{0\; \}\; F\; /\; v\_\{s\; \}\backslash right)^\{\backslash beta\}\backslash right)^\{1\; /\; \backslash beta\}\}$$

Here, $FF$ is the parallel electric field and ${\mu }_0\backslash mu\_0$ is the low-field mobility.

The saturation velocities are calculated by default from the temperature-dependent models^[3].

$$v_s=\frac{\alpha }{1+\theta \cdot exp\left(\frac{T_L}{T_{nom}}\right)}v\_s\; =\; \backslash frac\{\backslash alpha\}\{1+\backslash theta\; \backslash cdot\; exp\backslash left(\; \backslash frac\{T\_L\}\{T\_\{nom\}\}\; \backslash right)\}$$

The parameter $\beta \backslash beta$ depends on lattice temperature ($T_{L}T\_L$), and is given by:

$$\beta ={\beta }_0\cdot {\left(\frac{T_L}{300}\right)}^{{\beta }_{exp}}\backslash beta\; =\; \backslash beta\_0\; \backslash cdot\; \backslash left(\; \backslash frac\{T\_L\}\{300\}\; \backslash right)^\{\backslash beta\_\{exp\}\}$$

The parameters for this model with their default values (n-Si for electron values and p-Si for hole values) are defined in the following table:

Symbol	Parameter Name	Electron Value	Hole Value	Units
μ0	mu0	0.1450	0.0450	m2/(V*s)
β0	beta0	1.109	1.213	N/A
βexp	betaexp	0.66	0.17	N/A
α	alpha	2.4×105	2.4×105	m/s
θ	theta	0.8	0.8	N/A
Tnom	Tnom	600	600	K

The type of corresponding parameters are listed in the following table:

Symbol	Updated During Simulation	Predefinition of Material Parameter	Device Configuration	Description
F	✓	✗	✗	Parallel electric field (V/m)
μ0	✗	✓	✗	Low-field mobility (m2/(V*s))
β0	✗	✓	✗	N/A
βexp	✗	✓	✗	N/A
α	✗	✓	✗	N/A
θ	✗	✓	✗	N/A
Tnom	✗	✓	✗	N/A

Reference:

[1] Turin, Valentin O. "A modified transferred-electron high-field mobility model for GaN devices simulation." Solid-state electronics 49.10 (2005): 1678-1682. doi: 10.1016/j.sse.2005.09.002

[2] Canali, Claudio, et al. "Electron and hole drift velocity measurements in silicon and their empirical relation to electric field and temperature." IEEE Transactions on electron devices 22.11 (1975): 1045-1047. doi: 10.1109/T-ED.1975.18267

[3] Jacoboni, Canali, et al. "A review of some charge transport properties of silicon." Solid-State Electronics 20.2 (1977): 77-89. doi: 10.1016/0038-1101(77)90054-5

2.4 Transferred Electron Model

This model is used to calculate $\mu \backslash mu$.

The transferred electron model^{[1, 2]} is used in many III-V compound semiconductors which exhibit negative differential resistance.

The model is given by:

$$\mu =\frac{{\mu }_0+\left(v_s\mathrm{/}{F}_0\right){\left(F\mathrm{/}{F}_0\right)}^3}{1+{\left(F\mathrm{/}{F}_0\right)}^4}\backslash mu=\backslash frac\{\backslash mu\_\{0\}+\backslash left(v\_\{s\}\; /\; F\_\{0\}\backslash right)\backslash left(F\; /\; F\_\{0\}\backslash right)^\{3\}\}\{1+\backslash left(F\; /\; F\_\{0\}\backslash right)^\{4\}\}$$

where $FF$ is the parallel electric field, ${\mu }_0\backslash mu\_0$ is the low-field mobility, $F_{0}F\_\{0\}$ is a threshold field beyond which the carrier velocity saturates to a constant and $v_{s}v\_\{s\}$ is saturation velocity.

The saturation velocity is given by:

$$v_s=v_{s0}{\left(\frac{300}{T_L}\right)}^{\alpha }v\_s\; =\; v\_\{s0\}\; \backslash left(\; \backslash frac\{300\}\{T\_L\}\; \backslash right)^\{\backslash alpha\}$$

The parameters for this model with their default values (GaAs) are defined in the following table:

Symbol	Parameter Name	Electron Value	Units
μ0	mu0	0.75	m2/(V*s)
vs0	vs0	1×105	m/s
F0	F0	400000	V/m
α	alpha	2.3	N/A

The type of corresponding parameters are listed in the following table:

Symbol	Updated During Simulation	Predefinition of Material Parameter	Device Configuration	Description
F	✓	✗	✗	Parallel electric field (V/m)
μ0	✗	✓	✗	Low-field mobility (m2/(V*s))
vs0	✗	✓	✗	Saturation velocity (m/s)
F0	✗	✓	✗	N/A
α	✗	✓	✗	N/A
TL	✗	✗	✓	Lattice temperature (K)

Reference:

[1] Turin, Valentin O. "A modified transferred-electron high-field mobility model for GaN devices simulation." Solid-state electronics 49.10 (2005): 1678-1682. doi: 10.1016/j.sse.2005.09.002

[2] Selberherr, Siegfried. Analysis and simulation of semiconductor devices. Springer Science & Business Media, 1984.

2.5 Modified Transferred Electron Mode

This model is used to calculate $\mu \backslash mu$.

A modified transferred-electron mobility model designed for GaN devices^{[1, 2]}.

The total mobility is the combination of modified transferred electron model and Canali Model.

$$\mu ={\sin }^2(\pi \mathrm{/}2\cdot x)\cdot {\mu }_{MTE}(F)+{\cos }^2(\pi \mathrm{/}2\cdot x)\cdot {\mu }_C(F)\backslash mu=\backslash sin\; ^\{2\}(\backslash pi\; /\; 2\; \backslash cdot\; x)\; \backslash cdot\; \backslash mu\_\{MTE\}(F)+\backslash cos\; ^\{2\}(\backslash pi\; /\; 2\; \backslash cdot\; x)\; \backslash cdot\; \backslash mu\_\{C\}(F)$$

where ${\mu }_{MTE}\backslash mu\_\{MTE\}$ is given by:

$${\mu }_{MTE}=\frac{f(F)+(v_s\mathrm{/}{F}_{MT}){\left(F\mathrm{/}{F}_{MT}\right)}^{{\beta }_T-1}}{1+{\left(F\mathrm{/}{F}_{MT}\right)}^{{\beta }_T}}\backslash mu\_\{MTE\}=\backslash frac\{f(F)+(v\_\{\{s\}\}/F\_\{\{MT\}\})\backslash left(F\; /\; F\_\{\{MT\}\}\backslash right)^\{\backslash beta\_\{\{T\}\}-1\}\}\{1+\backslash left(F\; /\; F\_\{\{MT\}\}\backslash right)^\{\backslash beta\_\{\{T\}\}\}\}$$

where $F_{MT}F\_\{MT\}$ is a threshold field beyond which the carrier velocity saturates to a constant and $f(F)f(F)$ is a function as following:

$$f(F)=\frac{{\mu }_{low}}{{[1+{\left(\frac{{\mu }_{\text{low }}F}{({\mu }_{\text{low }}-{\mu }_{\text{high }})F_K+{\mu }_{\text{high }}F}\right)}^{{\beta }_K}]}^{1\mathrm{/}{\beta }_K}}f(F)=\backslash frac\{\backslash mu\_\{\{low\; \}\}\}\{\backslash left[1+\backslash left(\backslash frac\{\backslash mu\_\{\backslash text\; \{low\; \}\}\; F\}\{\backslash left(\backslash mu\_\{\backslash text\; \{low\; \}\}-\backslash mu\_\{\backslash text\; \{high\; \}\})\; F\_\{\{K\}\}+\backslash mu\_\{\backslash text\; \{high\; \}\}\; F\backslash right.\}\backslash right)^\{\backslash beta\_\{\{K\}\}\}\backslash right]^\{1\; /\; \backslash beta\_\{\{K\}\}\}\}$$

where $F_{K}F\_K$ is the kink electric field.

The Canali Model is given by:

$${\mu }_C(F)=\frac{{\mu }_{low}}{{(1+{\left({\mu }_{low}F\mathrm{/}{v}_s\right)}^{{\beta }_C})}^{1\mathrm{/}{\beta }_C}}\backslash mu\_\{C\}(F)=\backslash frac\{\backslash mu\_\{low\; \}\}\{\backslash left(1+\backslash left(\backslash mu\_\{low\; \}\; F\; /\; v\_\{s\; \}\backslash right)^\{\backslash beta\_\{C\}\}\backslash right)^\{1\; /\; \backslash beta\_\{C\}\}\}$$

The mixing parameter $xx$ is given by:

$$x=v_{MTE}\left(F_{MT}\right)\mathrm{/}{v}_{ref}x=v\_\{MTE\}\backslash left(F\_\{\{MT\}\}\backslash right)\; /\; v\_\{\{ref\}\}$$

The parameters for this model with their default values (GaN) are defined in the following table:

Symbol	Parameter Name	Electron Value	Units
μlow	mulow	0.1	m2/(V*s)
μhigh	muhigh	0.01	m2/(V*s)
vs	vs	1.91×105	m/s
vref	vref	2.86×105	m/s
FK	FK	1.4×106	V/m
FMT	FMT	2.57×107	V/m
βC	betaC	1.7	N/A
βT	betaT	5.7	N/A
βK	betaK	4	N/A

The type of corresponding parameters are listed in the following table:

Symbol	Updated During Simulation	Predefinition of Material Parameter	Device Configuration	Description
F	✓	✗	✗	Parallel electric field (V/m)
μlow	✗	✓	✗	N/A
μhigh	✗	✓	✗	N/A
vs	✗	✓	✗	Saturation velocity (m/s)
vref	✗	✓	✗	N/A
FK	✗	✓	✗	Kink electric field (V/m)
FMT	✗	✓	✗	N/A
βC	✗	✓	✗	N/A
βT	✗	✓	✗	N/A
βK	✗	✓	✗	N/A

Reference:

[1] Turin, Valentin O. "A modified transferred-electron high-field mobility model for GaN devices simulation." Solid-state electronics 49.10 (2005): 1678-1682. doi: 10.1016/j.sse.2005.09.002

[2] Farahmand, Maziar, et al. "Monte Carlo simulation of electron transport in the III-nitride wurtzite phase materials system: binaries and ternaries." IEEE Transactions on electron devices 48.3 (2001): 535-542. doi: 10.1109/16.906448

2.6 Poole-Frenkel Model

This model is used to calculate $\mu \backslash mu$.

Poole-Frenkel Model is the hopping mobility model commonly used for organic semiconductors.

The mobility from this model may be expressed using the following formula:

$$\mu ={\mu }_0\exp \left[{\left(F\mathrm{/}{F}_{cr}\right)}^{px}\right]\backslash mu=\backslash mu\_\{0\}\; \backslash exp\; \backslash left[\backslash left(F\; /\; F\_\{c\; r\}\backslash right)^\{p\; x\}\backslash right]$$

The type of corresponding parameters are listed in the following table:

Symbol	Updated During Simulation	Predefinition of Material Parameter	Device Configuration	Description
F	✓	✗	✗	Parallel electric field (V/m)
μ0	✗	✓	✗	N/A
Fcr	✗	✓	✗	N/A
px	✗	✓	✗	N/A

2.7 Tasch Model

This model is used to calculate $\mu \backslash mu$.

Tasch model^{[1, 2]} defines the mobility as a function of the perpendicular and parallel electric fields, the interface charge, the lattice temperature and the doping concentration and is given by the following expressions:

$$\mu =\mathrm{\Gamma }+(E_{\perp }-E_0)\cdot \frac{d\mathrm{\Gamma }}{d{E}_{\perp }}\backslash mu\; =\; \backslash Gamma\; +\; (E\_\{\backslash perp\}-E\_0)\; \backslash cdot\; \backslash frac\{d\backslash Gamma\}\{dE\_\{\backslash perp\}\}$$

where $E_{\perp }E\_\{\backslash perp\}$ is the transverse electric field and $E_{0}E\_0$ is the transverse electric field at the edge of the inversion layer.

The function $\mathrm{\Gamma }\backslash Gamma$ is defined as:

$$\mathrm{\Gamma }=\frac{{\mu }_{eff}}{(1+(\frac{{\mu }_{eff}\cdot E_{\parallel }}{v_{sat}}{)}^{\beta }{)}^{\frac{1}{\beta }}}\backslash Gamma\; =\; \backslash frac\{\backslash mu\_\{eff\}\}\{(1+(\backslash frac\{\backslash mu\_\{eff\}\; \backslash cdot\; E\_\{\backslash parallel\}\}\{v\_\{sat\}\})^\{\backslash beta\})^\{\backslash frac\{1\}\{\backslash beta\}\}\}$$

where $E_{\parallel }E\_\{\backslash parallel\}$ is the parallel electric field.

The carrier mobilities ${\mu }_{eff}\backslash mu\_\{eff\}$ are defined by three components ${\mu }_{ph}\backslash mu\_\{ph\}$, ${\mu }_{sr}\backslash mu\_\{sr\}$,and ${\mu }_c\backslash mu\_\{c\}$ that are combined by Mathiessen’s rule according to:

$${\mu }_{eff}={[\frac{1}{{\mu }_{ph}}+\frac{1}{{\mu }_{sr}}+\frac{1}{{\mu }_c}]}^{-1}\backslash mu\_\{e\; f\; f\}=\backslash left[\backslash frac\{1\}\{\backslash mu\_\{p\; h\}\}+\backslash frac\{1\}\{\backslash mu\_\{s\; r\}\}+\backslash frac\{1\}\{\backslash mu\_\{c\}\}\backslash right]^\{-1\}$$

The term ${\mu }_{ph}\backslash mu\_\{ph\}$ takes account of the effect of phonon scattering and is given by:

$${\mu }_{ph}^{-1}=({\mu }_b\cdot (\frac{T_L}{300}{)}^{-T_{{\mu }_b}}{)}^{-1}+(Z\mathrm{/}(D\cdot Y\cdot (\frac{T_L}{300}{)}^{0.5}){)}^{-1}\backslash mu\_\{ph\}^\{-1\}\; =\; (\backslash mu\_b\; \backslash cdot\; (\backslash frac\{T\_L\}\{300\})^\{-T\_\{\backslash mu\_b\}\})^\{-1\}\; +\; (Z/(D\backslash cdot\; Y\; \backslash cdot\; (\backslash frac\{T\_L\}\{300\})^\{0.5\}))^\{-1\}$$

The function $ZZ$ is defined as:

$$Z=Z_{11}\cdot (\frac{T_L}{300})\cdot E_{eff}^{-1}+Z_{22}\cdot E_{eff}^{-\frac{1}{3}}Z\; =\; Z\_\{11\}\backslash cdot(\backslash frac\{T\_L\}\{300\})\backslash cdot\; E\_\{eff\}^\{-1\}\; +\; Z\_\{22\}\backslash cdot\; E\_\{eff\}^\{-\backslash frac\{1\}\{3\}\}$$

where:

$$E_{eff}=\frac{E_{\perp }+(R-1)\cdot E_0}{R}E\_\{eff\}\; =\; \backslash frac\{E\_\{\backslash perp\}+(R-1)\backslash cdot\; E\_0\}\{R\}$$

The function $YY$ is defined as:

$$Y_n=P_1\cdot (\frac{T_L}{300}{)}^{B_1}+P_2\cdot n^{B_2}\cdot (\frac{T_L}{300}{)}^{-1}\cdot N_{f}Y\_n\; =\; P\_1\; \backslash cdot\; (\backslash frac\{T\_L\}\{300\})^\{B\_1\}\; +\; P\_2\; \backslash cdot\; n^\{B\_2\}\backslash cdot(\backslash frac\{T\_L\}\{300\})^\{-1\}\backslash cdot\; N\_f$$ $$Y_p=P_1\cdot (\frac{T_L}{300}{)}^{B_1}+P_2\cdot p^{B_2}\cdot (\frac{T_L}{300}{)}^{-1}\cdot N_{f}Y\_p\; =\; P\_1\; \backslash cdot\; (\backslash frac\{T\_L\}\{300\})^\{B\_1\}\; +\; P\_2\; \backslash cdot\; p^\{B\_2\}\backslash cdot(\backslash frac\{T\_L\}\{300\})^\{-1\}\backslash cdot\; N\_f$$

where $N_{f}N\_f$ is fixed interface charge per unit area at the gate dielectric-silicon interface.

The term ${\mu }_{sr}\backslash mu\_\{sr\}$ takes account of the effect of surface roughness and is calculated according to:

$${\mu }_{sr}=(\frac{E_{sr}}{E_{eff}}{)}^{\beta }\backslash mu\_\{sr\}\; =\; (\backslash frac\{E\_\{sr\}\}\{E\_\{eff\}\})^\{\backslash beta\}$$

The final term, ${\mu }_C\backslash mu\_C$, models Coulombic scattering with the expressions:

$${\mu }_{C,n}=\frac{N_2\ast (\frac{T_L}{T_0}{)}^{1.5}}{N_A\cdot [ln(1+{\gamma }_{BH,n})-\frac{{\gamma }_{BH,n}}{1+{\gamma }_{BH,n}}]}\backslash mu\_\{C,n\}\; =\; \backslash frac\{N\_2*(\backslash frac\{T\_L\}\{T\_0\})^\{1.5\}\}\{N\_A\; \backslash cdot\; [ln(1+\backslash gamma\_\{BH,\; n\})-\backslash frac\{\backslash gamma\_\{BH,\; n\}\}\{1+\backslash gamma\_\{BH,\; n\}\}]\}$$ $${\mu }_{C,p}=\frac{N_2\ast (\frac{T_L}{T_0}{)}^{1.5}}{N_D\cdot [ln(1+{\gamma }_{BH,p})-\frac{{\gamma }_{BH,p}}{1+{\gamma }_{BH,p}}]}\backslash mu\_\{C,p\}\; =\; \backslash frac\{N\_2*(\backslash frac\{T\_L\}\{T\_0\})^\{1.5\}\}\{N\_D\; \backslash cdot\; [ln(1+\backslash gamma\_\{BH,p\})-\backslash frac\{\backslash gamma\_\{BH,p\}\}\{1+\backslash gamma\_\{BH,p\}\}]\}$$

Here:

$${\gamma }_{BH,n}=\frac{N_1}{n}\cdot (\frac{T_L}{300}{)}^{\alpha }\backslash gamma\_\{BH,n\}\; =\; \backslash frac\{N\_1\}\{n\}\backslash cdot\; (\backslash frac\{T\_L\}\{300\})^\{\backslash alpha\}$$ $${\gamma }_{BH,p}=\frac{N_1}{p}\cdot (\frac{T_L}{300}{)}^{\alpha }\backslash gamma\_\{BH,p\}\; =\; \backslash frac\{N\_1\}\{p\}\backslash cdot\; (\backslash frac\{T\_L\}\{300\})^\{\backslash alpha\}$$

where $N_{A}N\_A$ is the channel acceptor doping concentration in $m^{-3}m^\{-3\}$, $N_{D}N\_D$ is the channel donor doping concentration in $m^{-3}m^\{-3\}$, $nn$ and $pp$ are the electron and hole concentrations per unit volume in the inversion layer ($m^{-3}m^\{-3\}$)

The parameters for this model with their default values (n-MOSFET for electron values and p-MOSFET for hole values) are defined in the following table:

Symbol	Parameter Name	Electron Value	Hole Value	Units
R	R	2	3	N/A
β	beta	2	1	N/A
μb	mub	0.115	0.0027	m2/(V*s)
Tμb	Tmub	2.5	1.4	N/A
D	D	3.2×10-9	2.35×10-9	N/A
P1	P1	0.09	0.334	N/A
B1	B1	1.75	1.5	N/A
P2	P2	4.53×10-8	3.14×10-7	N/A
B2	B2	-0.25	-0.3	N/A
Z11	Z11	0.0388	0.039	N/A
Z22	Z22	1.73×10-5	1.51×10-5	N/A
Esr	Esr	2.449×109	1.0×1010	V/m
N2	N2	1.1×1027	1.4×1024	m-3
N1	N1	2.0×1025	8.4×1022	m-3
α	alpha	2	3.4	N/A

The type of corresponding parameters are listed in the following table:

Symbol	Updated During Simulation	Predefinition of Material Parameter	Device Configuration	Description
E⊥	✓	✗	✗	Transverse electric field (V/m)
E0	✓	✗	✗	Transverse electric field at the interface (V/m)
E∥	✓	✗	✗	Parallel electric field (V/m)
R	✗	✓	✗	N/A
β	✗	✓	✗	N/A
μb	✗	✓	✗	N/A
Tμb	✗	✓	✗	N/A
D	✗	✓	✗	N/A
P1	✗	✓	✗	N/A
B1	✗	✓	✗	N/A
P2	✗	✓	✗	N/A
B2	✗	✓	✗	N/A
Z11	✗	✓	✗	N/A
Z22	✗	✓	✗	N/A
Esr	✗	✓	✗	N/A
N2	✗	✓	✗	N/A
N1	✗	✓	✗	N/A
α	✗	✓	✗	N/A
TL	✗	✗	✓	Lattice temperature (K)
NA	✗	✗	✓	Acceptor concentration (m-3)
ND	✗	✗	✓	Donor concentration (m-3)
n	✗	✗	✓	Electron concentration (m-3)
p	✗	✗	✓	Hole concentration (m-3)

Reference:

[1] Shin, Hyungsoon, et al. "A new approach to verify and derive a transverse-field-dependent mobility model for electrons in MOS inversion layers." IEEE transactions on electron devices 36.6 (1989): 1117-1124. doi: 10.1109/16.24356

[2] Shin, H., et al. "Physically-based models for effective mobility and local-field mobility of electrons in MOS inversion layers." Solid-State Electronics 34.6 (1991): 545-552. doi: 10.1016/0038-1101(91)90123-G

2.8 Yamaguchi Model

This model is used to calculate $\mu \backslash mu$.

The Yamaguchi Model^[1] consists of calculating the low-field, doping dependent mobility. Surface degradation is then accounted for based upon the transverse electric field before including the parallel electric field dependence.

The low-field part of the Yamaguchi Model is given as follows:

$${\mu }_0={\mu }_L\cdot [1+\frac{N}{\frac{N}{S}+N_{ref}}{]}^{-\frac{1}{2}}\backslash mu\_0\; =\; \backslash mu\_\{L\}\; \backslash cdot\; [1\; +\; \backslash frac\{N\}\{\backslash frac\{N\}\{S\}+N\_\{ref\}\}]^\{-\backslash frac\{1\}\{2\}\}$$

where $NN$ is the total impurity concentration. The equation parameters: ${\mu }_L\backslash mu\_L$, $SS$, $N_{ref}N\_\{ref\}$ are user-definable.

The transverse electric field dependence is accounted for as follows:

$${\mu }_s={\mu }_0\cdot (1+A_S\cdot E_{\perp }{)}^{-\frac{1}{2}}\backslash mu\_s\; =\; \backslash mu\_0\; \backslash cdot\; (1+A\_S\; \backslash cdot\; E\_\{\backslash perp\})^\{-\backslash frac\{1\}\{2\}\}$$

where $E_{\perp }E\_\{\backslash perp\}$ is the perpendicular electric field and the parameter, $A_{S}A\_\{S\}$, is user-definable.

The final calculation of mobility takes into account the parallel electric field dependence which takes the form:

$$\mu ={\mu }_s\cdot [1+{\left(\frac{{\mu }_s\cdot E_{\parallel }}{U_L}\right)}^2\cdot {(G+\frac{{\mu }_s\cdot E_{\parallel }}{U_L})}^{-1}+{\left(\frac{{\mu }_s\cdot E_{\parallel }}{V_S}\right)}^2{]}^{-\frac{1}{2}}\backslash mu\; =\; \backslash mu\_s\; \backslash cdot\; [1\; +\; \backslash left(\backslash frac\{\backslash mu\_s\; \backslash cdot\; E\_\{\backslash parallel\}\}\{U\_L\}\; \backslash right)^2\; \backslash cdot\; \backslash left(\; G\; +\; \backslash frac\{\backslash mu\_s\; \backslash cdot\; E\_\{\backslash parallel\}\}\{U\_L\}\backslash right)^\{-1\}+\; \backslash left(\; \backslash frac\{\backslash mu\_s\; \backslash cdot\; E\_\{\backslash parallel\}\; \}\{V\_S\}\; \backslash right)^2]^\{-\backslash frac\{1\}\{2\}\}$$

where $E_{\parallel }E\_\{\backslash parallel\}$ is the parallel electric field and the parameters: $U_{L}U\_\{L\}$, $V_{S}V\_\{S\}$ and $GG$ are user-definable.

The parameters for this model with their default values (n-MOSFET for electron values and p-MOSFET for hole values) are defined in the following table:

Symbol	Parameter Name	Electron Value	Hole Value	Units
S	S	350.0	81.0	N/A
Nref	Nref	3.0×1022	4.0×1022	m-3
μL	muL	0.14	0.0480	m2/(V*s)
AS	AS	1.54×10-7	5.35×10-7	m/V
VS	VS	1.036×105	1.2×105	m/s
UL	UL	4.9×104	2.928×104	m/s
G	G	8.8	1.6	N/A

The type of corresponding parameters are listed in the following table:

Symbol	Updated During Simulation	Predefinition of Material Parameter	Device Configuration	Description
E⊥	✓	✗	✗	Transverse electric field (V/m)
E∥	✓	✗	✗	Parallel electric field (V/m)
S	✗	✓	✗	N/A
Nref	✗	✓	✗	N/A
μL	✗	✓	✗	N/A
AS	✗	✓	✗	N/A
VS	✗	✓	✗	N/A
UL	✗	✓	✗	N/A
G	✗	✓	✗	N/A
N	✗	✗	✓	Total concentration (m-3)

Reference:

[1] Yamaguchi, Ken. "A mobility model for carriers in the MOS inversion layer." IEEE Transactions on Electron Devices 30.6 (1983): 658-663. doi: 10.1109/T-ED.1983.21185

Conclusion

In this article, we've delved into the key mobility models for electrons and holes in silicon (Si) semiconductors, exploring how they function under both low-field and high-field conditions. These models are not just theoretical constructs; they are practical tools that help engineers and researchers accurately simulate and optimize the behavior of semiconductor devices. The models discussed are indispensable for predicting carrier dynamics, accounting for factors such as velocity saturation and field-dependent mobility variations. Mastery of these models is essential for advancing semiconductor device performance and also enabling more accurate design processes.