C.1 Automatic Adjustment of the Channel Doping

To achieve a certain for a given device topography and source/drain doping the channel doping has to be adjusted accordingly. In modern deep-sub-micron devices the channel doping is determined by several parameters - depending on the type of channel engineering - so that the adjustment for is not unique. A practical and very robust approach is to define the channel doping relative to a certain channel doping level or relative to one specific implantation dose. This parameter is then varied to achieve the off-state current, while the other channel profile parameters (and so the shape of the profile) remain unchanged. The advantages of this procedure are that it is very robust and in an optimization it removes one unknown from the set of parameters but does not interfere in any other way.

Assuming a uniformly doped long-channel device in weak inversion the off-state current can be written as follows (cf. (A.14)):

$$\ensuremath{I_{\mathit{off}}}\xspace = A e^{-B\sqrt{\ensuremath{N_{\mathit{ch}}}\xspace }} .$$ (C.1)

From two simulations with different dopant profiles the parameters A and B can be determined as

$$\renewcommand {1.20}{1.6} \begin{array}{rcl} B &=& - \f... ..._1}-\sqrt{N_2}} \\ A &=& I_1 e^{B\sqrt{N_1}} . \end{array}$$ (C.2)

Of course, the assumptions above are violated by realistic devices. In practice, however, the dependence $\ensuremath{I_{\mathit{off}}}\xspace (\ensuremath{N_{\mathit{ch}}}\xspace )$ is still very close so that a next guess for can be computed from (C.1):

$$N = \left[-\frac{\ln(\ensuremath{I_{\mathit{off}}}\xspace /A)}{B}\right]^2 .$$ (C.3)

In the next step of the iteration I(N) is simulated and the worse of the pairs N₁,I₁ N₂,I₂ is replaced with I,N. This iteration is repeated from (C.2) until $-e_r < \log(I/\ensuremath{I_{\mathit{off}}}\xspace ) < +e_r$, where e_r is the acceptable error. Typically, the algorithm converges in 1-3 steps to <5% accuracy.