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The discretized semiconductor equations are represented by the nonlinear
equation system
f(x) = 0. The solution vector
x contains the carrier concentrations, the potential, the contact
voltages, and some more quantities depending on the simulation mode. As the
contact voltages are the quantities which determine the state of the device,
the dependence of the solution vector on the contact voltages is explicitly
written in the equation system. For Fig. 4.1 one gets
By applying Newton's algorithm to (4.3) with
V = V_{0} one gets
 ^{k}_{x}^{ . }u = f(x^{k}, V_{0}) with ^{k}_{x} =

(4.4) 
The iteration terminates after some convergence criterion is fulfilled. As a
next step the contact current
I(x, V_{0}) can be calculated. For the
calculation of the linearized conductance
G_{eq} the chain rule is
applied to (4.1) which gives
G_{eq} =
= ^{ . }
+ 
(4.5) 
where
I/x and
I/V are obtained by symbolic
differentiation of the function
I(x, V). The quantity
x/V can be determined in two different ways. The first
approach has been published in [76]. At first,
(4.4) is solved until a userdefined convergence criterion is
reached. Then a small perturbation V is applied on V_{0}. Using the
Taylor expansion of (4.3) with respect to both
x and V around the DC solution,
f(x_{0}, V_{0}) results
in
f(x_{0} + x, V_{0} + V)
f(x_{0}, V_{0}) + ^{ . }x + ^{ . }V +...= 0 
(4.6) 
As
f(x_{0}, V_{0}) = 0, the above equation reduces to
_{x}^{ . }x =  _{V}^{ . }V 
(4.7) 
The Jacobian matrix
_{x} is the same as for the last iteration
of the DC solution and the vector
_{V}^{ . }V on the
righthand side normally contains only a few nonzero elements as only the
quantities at the boundaries directly depend on V. The solution of the
linear system (4.7)
x is proportional to V, hence
x/V is independent of V and
equal to the limiting case which is the derivative when V becomes
infinitesimally small.
=
=  _{x}^{1 . }_{V} 
(4.8) 
Another equivalent possibility to calculate
x/V is from the
total derivative of (4.3) with respect to V which reads
and also results in (4.8). To summarize, the equivalent
conductance and current are calculated as follows
G^{k + 1}_{eq} 
= 
 ^{ . }_{x}^{1 . }_{V} +

(4.10) 
I^{k + 1}_{eq} 
= 
I^{k}  G^{k + 1}_{eq}^{ . }V^{k} 
(4.11) 
Next: 4.1.2 Modified TwoLevel Newton
Up: 4.1 The TwoLevel Newton
Previous: 4.1 The TwoLevel Newton
Tibor Grasser
19990531