Nope. Gummel method is better for converging. Newton method only works when your guess solution is close to real. Then it converges faster. But how do you know your solution closee to rela at first step? Impossible. Since these equations are nonlinear and hard to debug. The reason you don't get converge is that you don't have a proper damping method or you simply have a bug in the code.

You can download our solver in yrwu-wk.ee.ntu.edu.tw. All problem are solved by Gummel method. There is no big problem in converging.

I have no bugs, only problems when, for instance, the carrier lifetimes are too small, in the order of picoseconds, or when the deep trap density is higher than, let's say, 1e18 cm-3. Do you use damping in Gummel method? after the solution of each of the three equations?

Some damping might be needed. Also, for the trapping, if it is majority carrier, the recombination term needing carrier density should be moved into the matrix instead of leaving it in the RHS.

Convergence is usually slower for Gummel compared to full Newton. But Newton need to have good initial guess solution. Another important point is the Newton solver needs more memory and is more difficult to implement. As Yuh-Renn Wu said, for Gummel you need to implement some damping strategy.

I never used damping with Gummel, how does it work? Could you describe a simple example? Thanks!

With Gummel you solve Poisson, update potential, solve continuity for electrons with the previous potential, update n, solve continuity for holes with the previous potential and n, update p, and so on... Usually you need to damp the potential, n or p with a factor. And there are a lot of strategies depending on your problem. For example something like psi = psi / (1 + damp*psi).

I think there seems to be certain confusion, Gummel method is a strategy for solving the equations as described by Sid Hamady above. Newton method is the mathematical technique used to solve nonlinear equations: In the Gummel method you solve first Poisson's equation with the best available value for n and p, Newton method is one way to solve it? Thus in Gummel method (the Strategy) you are using Newton method (the technique). Similarly for the continuity equations.

Know we come to the point when the equations are strongly coupled, (could mean variables: potential, electron, and hole concentration are strongly dependent on each other like the case of semiconductors where n and p depends on the exponential of potential); In this case a small error in the potential could lead to values of n and p that are very far from the results which leads like wise to erroneous solution of the potential (damping is thus necessary).

In my opinion a better strategy is to solve all three equations simultaneously (needs large memory storage and maybe slower in each computation step) but your convergence would need lesser iteration steps and maybe you gain in the over all time.

Thank you all for your suggestions, I'll try damping with Gummel. How do you treat the recombination terms? Do you put them into the RHS or into the matrix? Could it be a critical issue? It seems to be complicated if you have the carrier density in the denominator of the R-G expression, so only under certain conditions it could be done, that's right?

Nacer Debbar is right: even in Gummel method, Newton is used to solve every subsystem (poisson, n and p) but usually we use 'Newton' term to speak about the full coupled method (solve simultaneously the whole system) and 'Gummel' to speak about the iterative decoupled method.

Recombination and generation term plays central role in the solving. Usually we update it after solving for Psi, n, p and use the updated R-G for the next iteration :

solve for Psi --> solve for p --> solve for n --> update R-G --> and son on...

Find attached my publication about numerical simulation of MSM PD. You might find explanation in the procedure I followed answers to your questions.

Article Investigation of the dark electrical characteristics of the ...

Thank you so much Nacer. I agree with Sidi about the denomination of Gummel method and Newton method to the decoupled and coupled ones, respectively.

Nacer, in order to solve the system (5) of your paper, you say that you solve it sequentially, right? so you solve Fpsi, then Fn and finally Fp. For solving Fn you use the last solution of Fpsi, and for solving Fp you use the last solution of Fpsi and Fn, and so on, that's right? It seems to be the Gummel decoupled method, using the Newton technique to solve each equation. Why do you say that "a better strategy is to solve all three equations simultaneously"? Please let me know if I am wrong.

Thanks for your time!

The system (4) is to be solved, you make an initial guess (Vo,Uo,Wo) then using Newton method to find the correction (dV,dU,dW), These corrections are solution to system (5) which is a large system of linear algebraic equation: (Number of equation is 3*number of points in your discretization). To solve it in one shot require the inversion of a very large matrix (very long time), but since these equations are linear all the coefficient are taken at the values (Vo,Uo,Wo), I preferred to iterate to find the solution it is much faster. This finishes one Newton step then you correct all the variables. Therefore at each Newton step you solve all the equation in system (4) together.