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B. Expressing the Equations (4.14)
In order to write (4.19) as a function of
the following set of rules is needed:
![\begin{displaymath}\begin{array}{ccc} (\bar{y}_{n}+y_{n})&=&X_{1}\quad, \\ (\bar{y}_{n}-y_{n})&=&X_{2}\quad, \end{array}\end{displaymath}](img986.png) |
(7.10) |
or
![\begin{displaymath}\begin{array}{ccc} \frac{X_{1}+X_{2}}{2}&=&y_{n}\quad, \\ \frac{X_{1}-X_{2}}{2}&=&\bar{y}_{n}\quad. \end{array}\end{displaymath}](img987.png) |
(7.11) |
The following identities (7.8):
![\begin{displaymath}\begin{array}(\bar{y}_{n}\pm y_{n})^{2}+\zeta^{2}&=&\left(1\p...
...{n}\, y_{n} &=& \frac{X_{1}^{2}-X_{2}^{2}}{4}\quad, \end{array}\end{displaymath}](img988.png) |
(7.12) |
allow to write
as function of
,
![\begin{displaymath}\begin{array}{ccc} \bar{y}_{n}^{2}+y_{n}^{2}+\zeta^{2}&=&1+\b...
...}&=&\frac{1-y_{n}^{2}-\zeta^{2}}{1-y_{n}^{2}}\quad, \end{array}\end{displaymath}](img989.png) |
(7.13) |
and
as function of
,
![\begin{displaymath}\begin{array}{ccc} \bar{y}_{n}^{2}+y_{n}^{2}+\zeta^{2}&=&1+\b...
...bar{y}_{n}^{2}-\zeta^{2}}{1-\bar{y}_{n}^{2}} \quad. \end{array}\end{displaymath}](img990.png) |
(7.14) |
Starting with the fraction of (4.20) in dimensionless form
, and putting in the definition of
,
leads to the term below:
![$\displaystyle I\equiv\frac{X_{2}\, \left(\zeta \pm \sqrt{\zeta^2 + X_{1}^{2}}\r...
...1}^{2}}\right)+X_{1}\, \left(\zeta \pm \sqrt{\zeta^2 + X_{2}^{2}}\right)}\quad.$](img992.png) |
(7.15) |
Proceeding by substituting
with (7.10) and
with (7.10), and using the identity:
![\begin{displaymath}\begin{array}{ccccc} \sqrt{X_{1,2}^{2}+\zeta^{2}}&=&\vert 1\p...
...2}}{4}\vert&=&\vert 1\pm\bar{y}_{n}y_{n}\vert\quad, \end{array}\end{displaymath}](img993.png) |
(7.16) |
the term (7.15) can be written as a function of
and
,
![$\displaystyle I\equiv\frac{\left(\bar{y}_{n}-y_{n}\right)\, \left( \zeta \pm \l...
..._{n}\right)\, \left( \zeta \pm \left(1 -\bar{y}_{n}y_{n}\right) \right) }\quad,$](img994.png) |
(7.17) |
Expanding the brakets and reorganizing the expression results in:
![$\displaystyle I\equiv -\frac{y_{n} \left(\zeta\pm 1 \mp \bar{y}_{n}^{2}\right)}{\bar{y}_{n} \left(\zeta \pm1 \mp y_{n}^{2}\right)}\quad.$](img995.png) |
(7.18) |
After substituting
for
the expression simplifies to:
![$\displaystyle I\equiv\frac{-y_{n}\zeta\left(1\pm\frac{\zeta}{1-y_{n}^{2}}\right...
...\pm1\mp y_{n}^{2}\right)\sqrt{\frac{1-y_{n}^{2}-\zeta^{2}}{1-y_{n}^{2}}}}\quad,$](img997.png) |
(7.19) |
and can be reformulated to the term used in (4.22):
![$\displaystyle I\equiv \mp \frac{y_{n}\, \zeta}{\sqrt{(1-y_{n}^{2})(1-y_{n}^{2}-\zeta^{2})}}\quad.$](img998.png) |
(7.20) |
Next: C. Estimating Diffusive Layer
Up: Appendix
Previous: A. Re-expressing as a
T. Windbacher: Engineering Gate Stacks for Field-Effect Transistors