Next: 4. Simulation Application
Up: 3.7 Generation and Recombination
Previous: 3.7.4 Band-to-Band Tunneling
Subsections
3.7.5 Impact Ionization
The impact ionization (II) models support both the drift-diffusion (DD) and the
hydrodynamic (HD) transport models, therefore, electric field dependent DD II models
and carrier temperature dependent HD II models are used in MINIMOS-NT.
3.7.5.1 Drift-Diffusion Impact Ionization
In DD simulation the model from [33] is used to calculate the II
generation rates for electrons and holes, respectively. The overall generation
rate is the sum of these two generation rates and can be expressed as a negative
recombination rate.
![\begin{displaymath}
-R^{\mathrm{II}}=G_{n}^{\mathrm{II}}+G_{p}^{\mathrm{II}}= \a...
...}} \cdot \frac{\left\vert\mathbf{J}_p\right\vert}{\mathrm{q}}.
\end{displaymath}](img693.gif) |
(3.152) |
The ionization coefficients
and
are expressed by Chynoweth's law
|
|
![$\displaystyle \alpha_{n,\mathrm {bulk}}=\alpha_{n,\mathrm {bulk}}^{\infty}\cdot...
...\mathbf{J}_n \right\vert}
{\mathbf{E}\cdot\mathbf{J}_n}\right)^{\beta_n}\right)$](img696.gif) |
(3.153) |
|
|
![$\displaystyle \alpha_{p,\mathrm {bulk}}=\alpha_{p,\mathrm {bulk}}^{\infty}\cdot...
...\mathbf{J}_p \right\vert}
{\mathbf{E}\cdot\mathbf{J}_p}\right)^{\beta_p}\right)$](img697.gif) |
(3.154) |
The default values are summarized in Table 3.38.
Table 3.38:
Parameter values for DD impact ionization model
Material |
[m ] |
[V/m] |
![$\beta_n$](img588.gif) |
[m ] |
[V/m] |
![$\beta_p$](img589.gif) |
Reference |
Si |
7.03e7 |
1.231e8 |
1.0 |
1.528e8 |
2.036e8 |
1.0 |
|
Ge |
1.55e9 |
1.560e8 |
1.0 |
1e9 |
1.28e8 |
1.0 |
[33] |
GaAs |
3.5e7 |
6.85e7 |
2.0 |
3.5e7 |
6.85e7 |
2.0 |
[190] |
GaP |
4.0e7 |
1.18e8 |
2.0 |
4.0e7 |
1.18e8 |
2.0 |
[190] |
|
To account for surface effects, the surface ionization rates
and
can deviate from the
bulk rates. The electron surface ionization rate is calculated in a similar
way (analog for holes)
![\begin{displaymath}
\alpha_{n,\mathrm {surf}}=\alpha_{n,\mathrm {surf}}^{\infty}...
...\vert}
{\mathbf{E} \cdot\mathbf{J}_n}\right)^{\beta_n}\right).
\end{displaymath}](img703.gif) |
(3.155) |
is given by (3.120) and depending on the
surface distance
describes a smooth transition between the surface and bulk
generation rates. The parameter
denotes a critical
length. The final surface dependent ionization rate
reads
![\begin{displaymath}
\alpha_{n,\mathrm {eff}}=F(y)\cdot \alpha_{n,\mathrm {surf}} + (1-F(y))\cdot\alpha_{n,\mathrm {bulk}}.
\end{displaymath}](img704.gif) |
(3.156) |
The effect is considered only for Si and the following values are used:
Table 3.39:
Parameter values for surface DD impact ionization model
Material |
[m ] |
[V/m] |
[m ] |
[V/m] |
[nm] |
Si |
1.03e7 |
1.50e8 |
4.0e8 |
3.0e8 |
10 |
|
In a HD simulation, the carrier temperatures are used as parameters
in the hydrodynamic impact ionization model. The implemented equation for the
electron generation rate depending on the concentration
and the bandgap
energy
[191,192] reads (analog for holes)
![\begin{displaymath}
G_n\left(T_n,n\right) = n\cdot A\cdot\left(\left(1+\frac{u}{...
...{2}\cdot\sqrt{u}\cdot
\exp\!\left(\frac{-1}{u}\right)\right),
\end{displaymath}](img707.gif) |
(3.157) |
![\begin{displaymath}
u = \frac{\mathrm{k_B}\cdot T_n}{E_{\mathrm{g}}}.
\end{displaymath}](img708.gif) |
(3.158) |
The prefactor
depends on the carrier and lattice temperatures and the local
bandgap
![\begin{displaymath}
A(T_{{\mathrm{L}}},T_n,E_{\mathrm{g}})=\frac{1}{C_1}\!\cdot ...
...\!C_4\!\cdot\! \frac{T_{{\mathrm{L}}}-T_0}{T_0}\right)\right).
\end{displaymath}](img710.gif) |
(3.159) |
The variables
and
correspond to 300 K and
,
respectively.
Table 3.40:
Parameter values for HD impact ionization model
Material |
[s] |
![$C_2$](img229.gif) |
![$C_3$](img635.gif) |
![$C_4$](img712.gif) |
Si |
9.531e-9 |
3.823 |
0.346333 |
0.0922 |
|
The overall generation rate is the sum of the electron and hole generation
rates, and is equal to a negative recombination rate
![\begin{displaymath}
R^{\mathrm{II}}=-G_{n}^{\mathrm{II}}-G_{p}^{\mathrm{II}}.
\end{displaymath}](img713.gif) |
(3.160) |
Another simple, but very practical model is available for modeling the impact
ionization rate in all semiconductors. It reads for electrons
![\begin{displaymath}
G_n\left(T_n,T_{{\mathrm{L}}},n\right) = n\cdot A\cdot\exp\!\left(\frac{-B\cdot E_{\mathrm{g}}}{\mathrm{k_B}\cdot T_n}\right)
\end{displaymath}](img714.gif) |
(3.161) |
and, respectively, for holes
![\begin{displaymath}
G_p\left(T_p,T_{{\mathrm{L}}},p\right) = p\cdot A\cdot\exp\!...
...t(\frac{-B\cdot E_{\mathrm{g}}}{\mathrm{k_B}\cdot T_p}\right).
\end{displaymath}](img715.gif) |
(3.162) |
The default values recommended for the simple HD II model are summarized in the
following table:
Table 3.41:
Parameter values for HD impact ionization model
Material |
[s ] |
![$B$](img716.gif) |
Si&Ge |
1e13 |
0.92 |
III-Vs |
1e13 |
1.0 |
|
This model has been already successfully applied in simulation of GaAs-based
and InP-based HEMTs [193,194]. However, it has not been
applied in simulation of III-V HBTs yet.
Next: 4. Simulation Application
Up: 3.7 Generation and Recombination
Previous: 3.7.4 Band-to-Band Tunneling
Vassil Palankovski
2001-02-28