Subsections

• 3.4.2.1 Mobility Model of MINIMOS 6
• 3.4.2.2 Model for Majority and Minority Electrons
• 3.4.2.3 Masetti Mobility Model

3.4.2 Ionized Impurity Scattering

In semiconductor devices mobility reduction due to ionized impurity scattering is a dominant effect. The influence of lattice and impurity scattering must be combined to obtain an effective mobility $\mu^{\mathrm{LI}}$. In MINIMOS-NT the following models are available.

3.4.2.1 Mobility Model of MINIMOS 6

To account for mobility reduction due to ionized impurity scattering, the formula of Caughey and Thomas [163] is used in conjunction with temperature dependent coefficients. $C_\mathrm {I}$ denotes the concentration of ionized impurities. The model is well applicable for Si.

$$\mu^{\mathrm{LI}}_{\nu} = \mu^{\mathrm{min}}_{\nu}+\frac{\mu^{\mathrm{L}}_{\nu}-\mu^{\mathr... ...yle 1+\left(\frac{C_\mathrm {I}}{C^{\mathrm{ref}}_{\nu}}\right)^{\alpha_{\nu}}}$$ (3.97)

$$\mu^{\mathrm{min}}_{\nu} = \left\{ \begin{array}{ll}\displaystyle \mu^{\mathrm{min}}_{\nu,30... ...\rm K}}\right)^{\gamma_{2,\nu}}&\hspace{5mm}T < 200 {\rm K} \end{array}\right.$$ (3.98)

$$C^{\mathrm{ref}}_{\nu} = C^{\mathrm{ref}}_{\nu,300}\cdot\left(\frac{T_{\mathrm{L}}}{\mathrm{300 K}}\right)^{\gamma_{3,\nu}}$$ (3.99)

$$\alpha_{\nu} = \alpha_{\nu,300}\cdot\left(\frac{T_{\mathrm{L}}}{\mathrm{300 K}}\right)^{\gamma_{4,\nu}}$$ (3.100)

Simpler expressions are applied to other basic materials. By setting $\gamma_{1,\nu}=\gamma_{2,\nu}$ and $\gamma_{4,\nu}=0$, (3.98) and (3.100), respectively, reduce to

$$\mu^{\mathrm{min}}_{\nu} = \mu^{\mathrm{min}}_{\nu,300}\cdot\left(\frac{T_{\mathrm{L}}}{\mathrm{300 K}}\right)^{\gamma_{1,\nu}}$$ (3.101)

$$\alpha_{\nu} = \alpha_{\nu,300}$$ (3.102)

The model parameter values are summarized in Table 3.22.

Table 3.22: Parameter values for the impurity mobility

Material	$\nu$	$\mu^{\mathrm{min}}_{\nu,300}$ [cm$^2$/Vs]	$\gamma_{1,\nu}$	$\gamma_{2,\nu}$	$C^{\mathrm{ref}}_{\nu,300}$ [cm$^{-3}$]	$\gamma_{3,\nu}$	$\alpha_{\nu,300}$	$\gamma_{4,\nu}$
Si	n	80	-0.45	-0.15	1.12e17	3.2	0.72	0.065
	p	45	-0.45	-0.15	2.23e17	3.2	0.72	0.065
Ge	n	850			2.6e17		0.56	0.0
	p	300			1.0e17		1.0	0.0
GaAs	n	800	-0.9	-0.9	1.0e17	6.2	0.5	0.0
	p	40			1.0e17	0.5	1.0	0.0
AlAs	n	10			1.0e17		0.5	0.0
	p	5			2.9e17	0.5	1.0	0.0
InAs	n	11700	-0.33	-0.33	4.4e16	3.6	0.5	0.0
	p	48			2.55e17	0.5	1.0	0.0
InP	n	1520	2.0	2.0	6.4e16	3.7	0.5	0.0
	p	24	1.2	1.2	2.5e17	0.47	1.0	0.0
GaP	n	76	-1.07	-1.07	2.85e17	1.8	0.5	0.0
	p	27			2.33e17		1.0	0.0

The results delivered by the model for the hole mobility as a function of the doping concentration for various III-V group binary semiconductors compared to measured data are shown in Fig. 3.33.

Figure 3.33: Hole mobility vs. doping concentration at 300 K: Comparison between the model and experimental data

3.4.2.2 Model for Majority and Minority Electrons

Though numerous theoretical and experimental papers [164,165,166] on electron mobility in semiconductors have been published there are still some issues under discussion, particularly in the very high doping regime. The difference between majority and minority electron mobility is a well-known phenomenon caused by effects such as degeneracy and the different screening behavior of electrons and holes in semiconductors. However, the mobility models usually employed in device modeling do not reflect these facts.

One of the basic assumptions in the models for ionized-impurity scattering is that the charge of an impurity center is treated as a point charge. In the approach from [140] it was shown that considering the spatial extent of the charge density one can explain various doping effects due to the chemical nature of the dopant at high doping concentrations. In [140] Monte-Carlo (MC) simulation results for the low-field mobility in silicon, covering arbitrary finite concentrations, temperatures and dopants are presented (see Fig. 3.34 and Fig. 3.35).

Figure 3.34: Majority mobility in P-, As- and Sb-doped silicon at 300 K: Comparison between MC simulation data and experimental data

Figure 3.35: Minority mobility in B-doped silicon as a function of concentration: MC simulation data at different temperatures

The minority mobility at doping levels above $10^{19}$ cm$^{-3}$ exceeds the majority mobility more than three times at 300 K. The difference gets even stronger at low temperatures (up to sixteen times).

A model which distinguishes between the majority and minority electrons in $\mathrm{Si}$, as well as between dopant species is described in [167,168,66]. Although initially proposed for the majority electron mobility in Si

$$\mu^{\mathrm{LI}}_{n} = \frac{\mu^{\mathrm{L}}_n - \mu_1 - \... ...{\left(\frac{C_{\mathrm{I}}}{C_{2}}\right)}^ {\beta}}} + \mu_2$$ (3.103)

offers enough flexibility to model also the minority electron mobility in Si (see Fig. 3.36). In general, it can be applied also for any other material of interest (Fig. 3.37, Fig. 3.38). (3.103) is similar to (3.97), a function with two extreme values ( $\mu^{\mathrm{L}}$ as a maximum and $\mu_1$ as a minimum mobility). (3.103) is a mathematical function which can deliver a second maximum or minimum at very high impurity concentrations depending on the sign of $\mu_1$. Thus, it allows both majority and minority carrier mobilities to be properly modeled.

Figure 3.36: Comparison of the analytical model and MC data for electron mobility in Si at 300 K

Figure 3.37: Comparison of the analytical model and MC data for electron mobility in InP at 300 K

Figure 3.38: Comparison of the analytical model and MC data for electron mobility in GaAs at 300 K

In [169] an automated parameter extraction using an optimizer [170] for the mobility models was presented. Most of the existing experimental data on the low-field mobility together with accurate MC simulations for Si [140] and for III-V semiconductor compounds [128,171,172] are used as input. The temperature dependence of the lattice mobility $\mu^{\mathrm{L}}_n$ preserves the expression (3.96). The majority electron mobility $\mu^{\mathrm{maj}}_{n}$ is modeled as a function of the donor concentration $N_\mathrm {D}$ and the lattice temperature. The temperature dependence of the parameters $\mu_1$, $\mu_2$, $C_1$, $C_2$, $\alpha$, and $\beta$ is modeled by simple power laws.

$$\mu^{\mathrm{maj}}_{n} = \frac{\mu^{\mathrm{L}}_n - \mu_1 - \mu_2} {\displaystyle{1+ {\lef... ...isplaystyle{1+ {\left(\frac{ N_{\mathrm {D}}}{C_{2}}\right)}^ {\beta}}} + \mu_2$$ (3.104)

$$\mu_1 = \mu^{\mathrm{maj}}_{1,300}\cdot\left(\frac{T_{\mathrm{L}}}{\mathrm{300 K}}\right)^{\gamma_{1}}$$ (3.105)

$$\mu_2 = \mu^{\mathrm{maj}}_{2,300}\cdot\left(\frac{T_{\mathrm{L}}}{\mathrm{300 K}}\right)^{\gamma_{2}}$$ (3.106)

$$\alpha = \alpha_{300}\cdot\left(\frac{T_{\mathrm{L}}}{\mathrm{300 K}}\right)^{\gamma_{3}}$$ (3.107)

$$\beta = \beta^\mathrm {maj}_{300}\cdot\left(\frac{T_{\mathrm{L}}}{\mathrm{300 K}}\right)^{\gamma_{4}}$$ (3.108)

$$C_1 = C_{1,300}\cdot\left(\frac{T_{\mathrm{L}}}{\mathrm{300 K}}\right)^{\gamma_{5}}$$ (3.109)

$$C_2 = C^\mathrm {maj}_{2,300}\cdot\left(\frac{T_{\mathrm{L}}}{\mathrm{300 K}}\right)^{\gamma_{6}}$$ (3.110)

The minority electron mobility $\mu^{\mathrm{min}}_{n}$ is modeled as a function of the acceptor concentration $N_\mathrm {A}$ and the lattice temperature. The parameters $C_1$ and $\alpha$ preserve the values from (3.108) and (3.110), respectively. The new parameters $\mu_1$, $\mu_2$, $C_2$, and $\beta$, used in the calculation of $\mu^{\mathrm{min}}_{n}$, again follow simple power laws as a function of temperature.

$$\mu^{\mathrm{min}}_{n} = \frac{\mu^{\mathrm{L}}_n - \mu_1 - \mu_2} {\displaystyle{1+ {\lef... ...isplaystyle{1+ {\left(\frac{ N_{\mathrm {A}}}{C_{2}}\right)}^ {\beta}}} + \mu_2$$ (3.111)

$$\mu_1 = \mu^{\mathrm{min}}_{1,300}\cdot\left(\frac{T_{\mathrm{L}}}{\mathrm{300 K}}\right)^{\gamma_{7}}$$ (3.112)

$$\mu_2 = \mu^{\mathrm{min}}_{2,300}\cdot\left(\frac{T_{\mathrm{L}}}{\mathrm{300 K}}\right)^{\gamma_{8}}$$ (3.113)

$$\beta = \beta^\mathrm {min}_{300}\cdot\left(\frac{T_{\mathrm{L}}}{\mathrm{300 K}}\right)^{\gamma_{9}}$$ (3.114)

$$C_2 = C^\mathrm {min}_{2,300}\cdot\left(\frac{T_{\mathrm{L}}}{\mathrm{300 K}}\right)^{\gamma_{10}}$$ (3.115)

To account for a superposition of doping profiles a harmonic mean is used [168].

$$\mu^{\mathrm{LI}}_n = \left(\frac{1}{\mu^{\mathrm{maj}}_n} +... ...^{\mathrm{min}}_n} - \frac{1}{\mu^{\mathrm{L}}_n}\right)^{-1}$$ (3.116)

The model parameters used for Si in the range 70-500 K and for GaAs and InP at 300 K are summarized in Table 3.23.

Table 3.23: Parameter values for the majority/minority impurity mobility

Parameter	Si	GaAs	InP	Unit
$\mu^{\mathrm{maj}}_{1,300}$	52	2590	1140	$\mathrm{ cm^2/Vs}$
$\gamma_{1}$	-0.18	-	-
$\mu^{\mathrm{maj}}_{2,300}$	8	133	20	$\mathrm{ cm^2/Vs}$
$\gamma_{2}$	-1.49	-	-
$\mu^{\mathrm{min}}_{1,300}$	-200	-750	-742	$\mathrm{ cm^2/Vs}$
$\gamma_{7}$	-0.58	-	-
$\mu^{\mathrm{min}}_{2,300}$	230	1400	1920	$\mathrm{ cm^2/Vs}$
$\gamma_{8}$	-1.02	-	-
$\alpha_{300}$	0.7	0.7	0.6
$\gamma_{3}$	0.02	-	-
$\beta^\mathrm {maj}_{300}$	5.33	1.7	2.5
$\gamma_{4}$	-9.5	-	-
$\beta^\mathrm {min}_{300}$	2.0	2.8	3.2
$\gamma_{9}$	-1.2	-	-
$C_{1,300}$	1.17e17	0.5e17	4e16	$\mathrm{ cm^{-3}}$
$\gamma_{5}$	3.55	-	-
$C^\mathrm {maj}_{2,300}$	5.8e20	1.8e19	1.6e19	$\mathrm{ cm^{-3}}$
$\gamma_{6}$	0.134	-	-
$C^\mathrm {min}_{2,300}$	1.0e19	1.4e19	1.6e19	$\mathrm{ cm^{-3}}$
$\gamma_{10}$	0.12	-	-

3.4.2.3 Masetti Mobility Model

The Masetti bulk mobility model [173] is a default mobility model in several device simulators, e.g. [40,41]. It can be treated as a simple case of (3.97), only valid at 300 K.

$$\mu^{\mathrm{LI}}_{\nu} = \mu^{\mathrm{}}_\mathrm {min1}\cdo... ...hrm{}}_1}{\displaystyle{1+\left(\frac{C_s}{C_I}\right)^\beta}}$$ (3.117)

It can be useful to compare some of the parameters from the majority mobility model at 300 K to the default values of the Masetti model (without $P_c=0$) as it is shown in Table 3.24.

Table 3.24: Comparison between model parameters for majority electrons in Si at 300 K and the parameter values for the Masetti impurity mobility model

Parameter	Si	Parameter	Masetti	Unit
$\mu^{\mathrm{maj}}_{1,300}$	52	$-\mu^{\mathrm{}}_1$	-56.1	$\mathrm{ cm^2/Vs}$
$\mu^{\mathrm{maj}}_{2,300}$	8	$\mu^{\mathrm{}}_\mathrm {min1}$	68.5	$\mathrm{ cm^2/Vs}$
$\mu^{\mathrm{maj}}_{1,300}+\mu^{\mathrm{maj}}_{2,300}$	60	$\mu^{\mathrm{}}_\mathrm {min2}$	68.5	$\mathrm{ cm^2/Vs}$
$C_{1,300}$	1.17e17	$C_r$	9.2e16	$\mathrm{ cm^{-3}}$
$C^\mathrm {maj}_{2,300}$	5.8e20	$C_s$	3.4e20	$\mathrm{ cm^{-3}}$
$\alpha$	0.7	$\alpha$	0.711
$\beta$	5.33	$-\beta$	-1.98