3.2.2 Discretization of the Basic Equations and Numerical Aspect



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3.2.2 Discretization of the Basic Equations and Numerical Aspect

 

The strongly implicit discretization of the basic semiconductor equations 3.1, 3.2 and 3.3 is often applied to solve the transient problem in semiconductor devices. The difference system obtained by the strongly implicit discretization, linearized at an equilibrium point, is absolutely stable (stable independent of time step)gif. The difference equations are nonlinear and coupled. Thereby, they ought to be solved simultaneously, using e.g. Newton's iterative method. This amounts to solving a large system of equations, instead of solving three much smaller systems as in a sequential approach. Moreover, when including the trap-dynamics equations 3.11, the number of space variables becomes very large in a simultaneous approach.

Therefore, we will apply a sequential algorithm to solve the basic semiconductor equations 3.1, 3.2 and 3.3, including the trap-dynamics equations 3.11. The discretization methods and schemes for solving the corresponding difference equations in a sequential manner have been considerably investigated for the semiconductor-device problems in the literature [373][314][313][283], but usually assuming vanishing generation-recombination. Some of these results which we directly adapted to solve our problem are repeated here. It is worth to note that the presented algorithm has shown absolute stability and high efficiency in all charge-pumping simulations carried out for different MOS devices and a variety of the terminal pulses.

Let us assume that the solution is known for the moment : the potential , carrier concentrations and and the trap occupancy function . We consider the solution for the time step . After integrating equation 3.1 in the interval and multiplying by we arrive at

 

Since the explicit discretization leads to difference equations which are not stable for large time steps ([440]), we prefer to discretize both time-dependent current-continuity equations 3.1 and 3.2 by the strongly implicit method with respect to carrier concentrations. After replacing and with their values for , and it follows

 

 

The discretized Poisson equation 3.3 reads

 

In the previous equations and henceforward we drop the discretization in the position space. Let us neglect the generation-recombination terms at present. M.Sever (his previous family name was Mock) has proven convergence ([440]) of the difference equations 3.13 - 3.15 in [314]. He also claimed this scheme is not stable for the time steps larger than the minimal effective dielectric time constant in device . This time constant is very small for common parameters, much shorter than the time intervals involved in the charge pumping.
To overcome the instability of 3.13 - 3.15 M.Sever has proposed in [314][313] an unconditionally stable difference scheme. It is based on solving the total current continuity equation

 

instead of the Poisson equation 3.3. In 3.16, is the displacement current. Expression 3.16 follows by replacing 3.1 and 3.2 into 3.3 after differentiating 3.3 in time. The trap-related terms vanish after the substitution, because holds. This scheme consists of the linear difference equations 3.13, 3.14 and

 

which is a nonlinear difference equation related to the continuity equation for the total current [373][313]. An other proposal for this method is presented in [314] (see also [416]). It consists of equation 3.17 and

 

 

Equations 3.18 and 3.19 are the same as 3.13 and 3.14, but and are evaluated using new potential instead of , after solving 3.17 first. Both systems of difference equations are absolutely stable, linearized at the equilibrium assuming vanishing generation-recombination [314][313]. For both systems, the solution does not satisfy the Poisson equation at , as it is evident after replacing the solution in 3.15 gif. However, based on this discretization method, M.Sever has constructed an iterative scheme which exhibits very good behavior [314]:

 

 

 

where is an iteration counter. The procedure consists of sequentially solving 3.20, 3.21 and 3.22. At the beginning , and are assumed. After [314], this procedure linearized at the equilibrium assuming vanishing generation-recombination converges independent of the time step . For (one sequence) it is equivalent to 3.17 - 3.19. For the solution of the procedure 3.20 - 3.22 converges to the solution of the nonlinear coupled system obtained by the strongly implicit discretization of the basic semiconductor equations 3.1, 3.2 and 3.3 ([314]), thus it satisfies the Poisson equation. In practice, after truncating the procedure at a finite , the Poisson equation is satisfied with some residual error.
To reduce the residual of the Poisson equation B.S.Polsky and J.S.Rimshans developed in [373] (and further applied in [283]) an absolute stable extension of the difference scheme 3.13, 3.14 and 3.17. They solve this system with the Poisson equation 3.15 extended with a conveniently chosen damping (stabilizing) term. The potential from 3.17 is applied as a predictor for the Poisson equation to obtain , which results in a remarkable reduction of the residual error in the Poisson equation (see footnote 17). However, according to [373][283] it seems that they did not apply the outstandingly good iterative algorithm 3.20 - 3.22.

In the algorithm we employ, the sequential procedure 3.20, 3.22 and 3.21 is applied at the beginning of every new time step. After a given accuracy is achieved, the time-dependent Gummel procedure ([159]) is applied to obtain the final solution, using the solution of 3.20, 3.22 and 3.21 as an initial solution. For the `switch criterion' we chose that the maximal local correction in the potential becomes smaller than the assumed value (Figure 1.2 in [191]). The time-dependent Gummel procedure we apply follows from 3.13, 3.14 and 3.15:

 

 

 

where is the iteration counter. Additional details of the algorithm are given in [191]. Since the final solution is obtained by solving 3.25, the Poisson equation is fulfilled with the desired (controlled) accuracy at , independent of the time step. No proof for the convergence of the algorithm is available now. From the practical side, however, this algorithm has always shown absolute and fast convergence for time steps ranged from very short () to very long () in many simulations. It combines the stability of the initial procedure 3.20, 3.22 and 3.21 with faster convergence of the final procedure 3.23, 3.24 and 3.25.

For further discussion the difference equation 3.24 is written discretized in both time and position space

 

where the matrix , due to discretization in the position space, depends either on or . is the unit matrix. is the net generation rate averaged in time (the last right-hand-side term in 3.24). Since high trap densities commonly occur in practice, for both interface and bulk traps, the term can introduce a large local perturbation in the carrier continuity equation. Consequently, this term is evaluated here in advance for . In deriving 3.26 we used

 

The same idea is applied in the discretization of the Poisson equation 3.25 in time and position space (see Chapter 7 in [416] also)

 

where is the difference form of the Laplace operator . The term at the main diagonal of the left-hand-side matrix is crucial for a stable and efficient convergence of the sequential algorithm presented, when high trap densities are assumed in device.



next up previous contents
Next: 3.2.3 Solution of the Up: 3.2 Physical Model and Previous: 3.2.1 Physical Model



Martin Stiftinger
Sat Oct 15 22:05:10 MET 1994