B. Driving Force Discretization

TO IMPLEMENT the discretization scheme described in Chapter 3 into the device simulator MINIMOS-NT an expression for the driving force was required. The driving force $ \ensuremath{\boldsymbol{\mathrm{F}}}_n$ is defined by

$\displaystyle \ensuremath{\boldsymbol{\mathrm{J}}}_n = \mathrm{q}\, \mu_n \, n \, \ensuremath{\boldsymbol{\mathrm{F}}}_n \ .$ (B.1)

To obtain the discrete driving force the discretized current density eqns. (3.63) and (3.64)

$\displaystyle J_n$ $\displaystyle = - \frac{C_1}{\Delta x} \, \ensuremath{\overline{T_n}}\, \Bigl( ...
.....}\ensuremath{\overline{T_n}}= \frac{\Delta T_n}{\ln(T_{n \, j} / T_{n \, i})}$ (B.2)
$\displaystyle Y_1$ $\displaystyle = - \frac{1}{\ensuremath{\overline{T_n}}} \, \Bigl( s_n \, \frac{\mathrm{q}}{\mathrm{k}_\mathrm{B}} \, \Delta \psi + \Delta T_n \Bigr) \ ,$ (B.3)

must therefore be divided in some way by the electron concentration $ n$. Thus, an average carrier concentration $ \ensuremath{\overline{n}}$ is introduced via the following definition

$\displaystyle J_n \stackrel{\textstyle !}{=} - \frac{C_1}{\Delta x} \, \ensurem...
...th{\overline{n}}\, \ensuremath{\mathcal{B} \left( - \Lambda \right) }\Bigr) \ .$ (B.4)

By comparing the coefficients of eqn. (B.4) with those from eqn. (B.2)

$\displaystyle \ensuremath{\overline{n}}\, \ensuremath{\mathcal{B} \left( \Lambda \right) }$ $\displaystyle = n_j \, \ensuremath{\mathcal{B} \left( Y_1 \right) }\ ,$ (B.5)
$\displaystyle \ensuremath{\overline{n}}\, \ensuremath{\mathcal{B} \left( - \Lambda \right) }$ $\displaystyle = n_i \, \ensuremath{\mathcal{B} \left( - Y_1 \right) }\ ,$ (B.6)

and using the identity

$\displaystyle \frac{\ensuremath{\mathcal{B} \left( x \right) }}{\ensuremath{\mathcal{B} \left( -x \right) }} = e^{-x} \ ,$ (B.7)

the new argument $ \Lambda$ of the BERNOULLI function can be calculated

$\displaystyle e^{ -\Lambda}$ $\displaystyle = \frac{n_j}{n_i} \, e^{- Y_1} \ ,$ (B.8)
$\displaystyle \Lambda$ $\displaystyle = Y_1 - \ln(n_j / n_i) \ ,$ (B.9)

and the average carrier concentration is finally found to be

$\displaystyle \textcolor{lightgrey}{.......}\ensuremath{\overline{n}}= n_j \, \...
...\left( - Y_1 \right) }}{\ensuremath{\mathcal{B} \left( - \Lambda \right) }} \ .$ (B.10)

Applying the identity

$\displaystyle \ensuremath{\mathcal{B} \left( x \right) }- \ensuremath{\mathcal{B} \left( -x \right) }= -x \ ,$ (B.11)

to eqn. (B.4) yields

$\displaystyle J_n = - \frac{C_1}{\Delta x} \, \ensuremath{\overline{T_n}}\, \en...
...lta x} \, \ensuremath{\overline{T_n}}\, \ensuremath{\overline{n}}\, \Lambda \ .$ (B.12)

After inserting $ C_1$ from eqn. (2.188)

$\displaystyle J_n = \frac{s_n \, \mathrm{k}_\mathrm{B}\, \mu_n}{\Delta x} \, \e...
...{B}}{\mathrm{q}} \, \ensuremath{\overline{T_n}}\, \frac{\Lambda} {\Delta x} \ ,$ (B.13)

the expression for the discretized driving force can easily be obtained

$\displaystyle \boxed{ F_n = s_n \, \frac{\mathrm{k}_\mathrm{B}}{\mathrm{q}} \, ...
..._n}{\ln(T_{n \, j} / T_{n \, i})} \, \frac{Y_1 - \ln(n_j / n_i)}{\Delta x} }\ .$ (B.14)

The consistency of the discretization can be checked by calculating the driving force in the limit of $ \Delta x \to 0$

$\displaystyle \lim_{\Delta x \to 0} F_n$ $\displaystyle = \lim_{\Delta x \to 0} \, s_n \, \frac{\mathrm{k}_\mathrm{B}}{\m...
...\Delta x} - \frac{\ensuremath{\overline{T_n}}\, \Delta \ln(n)}{\Delta x} \Bigr)$ (B.15)
  $\displaystyle = \lim_{\Delta x \to 0} \, s_n \, \frac{\mathrm{k}_\mathrm{B}}{\m...
...\frac{\Delta T_n}{\Delta \ln(T_n)} \, \frac{\Delta \ln(n)}{\Delta x} \Bigr) \ ,$ (B.16)

where the abbreviations for $ \Lambda$ and $ Y_1$ have been expanded. Using the total derivative yields

$\displaystyle F_n$ $\displaystyle = - \ensuremath{\frac{\ensuremath{\mathrm{d}}\psi}{\ensuremath{\m...
...nsuremath{\frac{\ensuremath{\mathrm{d}}\ln(n)}{\ensuremath{\mathrm{d}}x}}\Bigr)$ (B.17)
  $\displaystyle = E - s_n \, \frac{\mathrm{k}_\mathrm{B}}{\mathrm{q}} \, \frac{1}...
...\, \ensuremath{\frac{\ensuremath{\mathrm{d}}n}{\ensuremath{\mathrm{d}}x}}\Bigr)$ (B.18)
  $\displaystyle = E - s_n \, \frac{\mathrm{k}_\mathrm{B}}{\mathrm{q}} \, \frac{1}...
...suremath{\frac{\ensuremath{\mathrm{d}}(n \, T_n)}{\ensuremath{\mathrm{d}}x}}\ ,$ (B.19)

which is the one-dimensional projection of the driving force

$\displaystyle \ensuremath{\boldsymbol{\mathrm{F}}}_n = \ensuremath{\boldsymbol{...
...\frac{1}{\mathrm{q}\, \mu_n \, n} \, \ensuremath{\boldsymbol{\mathrm{J}}}_n \ .$ (B.20)

M. Gritsch: Numerical Modeling of Silicon-on-Insulator MOSFETs PDF