This section gives an overview about the steady-state and transient simulation modes including a discussion of the nonlinear solution technique. For the steady-state analysis, the discretized equations (2.21), (2.22), and (2.23) can be symbolically written as:

with

(2.102) |

Note that for the sake of simplification, the vectors of the discretized
quantities and equations are not explicitly noted. The resulting discretized
problem is then usually solved by a damped *Newton* method which requires the
solution of a linear equation system at each step. The result of the
steady-state simulation mode is the operating point, which is a prerequisite
for any subsequent transient or small-signal simulation.

2.3.1 Solving the Nonlinear System

As the resulting discretized equation system is still nonlinear, the solution , which is assumed to exist is obtained by applying a linearization technique. The nonlinear problem can be defined as

(2.103) |

with

(2.104) |

Most iterative methods are based on a fixpoint equation , where is constructed in such a way that the fixpoint is a solution of that equation [193]. During the iteration the error between the current solution of the -th iteration step and converges to zero, if specific properties and requirements on the initial guess are fulfilled. With a neighborhood , , and a constant , the iteration will converge for any to , if

Then,
is a so-called contractive mapping, and the locally
convergent iteration does converge for any
to
. In order to
fulfill (2.105) it is assumed that the *Frechet*
derivative
exists at the fixpoint
and that its eigenvalues
are less than one in modulus [193]. According to the
*Ostrowski* theorem [243],
is contractive if
the spectral radius
, which is the maximal modulus of
all eigenvalues of
. If
exists such that

(2.106) |

is the

(2.107) |

where is the

As the iteration can be rewritten in the form

(2.109) |

the

It is important to note that
must only be an approximation of the
*Frechet* derivative, which follows from the derivation of
[193]. Furthermore, in order to enlarge the radius of convergence and
thus improve the convergence behavior of the *Newton* approximation,
the couplings between the equations can be reduced, especially during the first
steps of the iteration. Before the update norm, that is the infinity norm of
the update vectors of all quantities, has fallen below a specified value, the
derivatives as shown in Table 2.1 are normally ignored. Besides the
driving force for electrons and holes in the drift-diffusion model, and
, and the tunneling current density
, all quantities are already
known from the previous sections. Note that for the sake of simplification just
the symbols are given without vector notations.

The linear equation system for the -th iteration step looks like:

(2.110) |

The right-hand-side vector is the residual and is the update and correction vector. This solution vector of the linear equation system is used to calculate the next solution of the

(2.111) |

To avoid overshoot of the solution and to extend the local convergence of the method several damping schemes suggested by

(2.112) |

Investigations have shown that damping based on the potential delivers the most promising results [65]:

where is an adjustable parameter of the damping scheme, the update norm of the potential sub-vector, and the thermal voltage. Larger yields more logarithm-like damping. The potential damping scheme avoids the expensive evaluation of the right-hand-side vector, which is for example required for the scheme of

2.3.2 Transient Simulation

The transient problem arises if the boundary condition for the electrostatic potential or the contact currents becomes time-dependent. Hence, the partial time derivatives of the carrier concentrations in (2.22) and (2.23) have to be taken into account.

There are several approaches for transient analysis [193], among them
are the forward and backward *Euler* approaches. Whereas the former
shows significant stability problems, the latter is unconditionally stable for
arbitrarily large time steps .
However, full backward time differencing
requires much computational resources for solving the large nonlinear
equation system at each time step, but gives good results.
The quality of the results can be measured by the truncation error
[146]. Equations (2.21), (2.22) and
(2.23), discretized in time and symbolically written, read then at the
-th time step when is to be calculated:

(2.114) | |

(2.115) | |

(2.116) |

From a computational point of view it is to note, that in comparison to the steady-state solution the algebraic equations arising from the time discretization are significantly easier to solve [193]. This has mainly two reasons: first, the partial time derivatives help to stabilize the spatial discretization. Second, the solutions can be used as a good initial guess for the next time step. Furthermore, the equation assembly structures can be reused (see Section 4.12).

S. Wagner: Small-Signal Device and Circuit Simulation