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George Pau @ on
convergence criteria
In the input deck, the poisson convergence criterion is stated as 1e-3. What exactly is the convergence criteria used in omenwire? How much additional computational time will I need if I decrease the tolerance by a factor of 10?
Thanks, George
SungGeun Kim @ on
Hi George,
Thank you for the question. For now, the convergence criterion used in omenwire is 1e-3. The next version of omenwire will allow users to change the criterion. The additional computing time, as you decrease it by a factor of 10, depends on the structure of nanowire and the environment which users setup. I will check this out and come back to you shortly, but reasonable explanation without running simulations with different criterion could be given after a while.
Thank you
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Mathieu Luisier @ on
The Poisson equation is solved using a Newton-Raphson scheme in OMEN. There is an outer loop where the Schrodinger and Poisson equations are self-consistently solved and an inner loop in the Poisson equation where the carrier density is updated using simple analytical formula.
The convergence criterion that you currently see in the input deck is the criterion of the outer loop. It means that between two iterations, the normalized norm of the electrostatic potential should not vary by more than 1e-3. It is difficult to say how much longer the simulation will last if you set this criterion to 1e-4. The number of additional iterations that are required depends on the nanowire structure and on the bias conditions, as the time per iteration.
The convergence criterion of the inner loop can also be changed. The default value is 1e-3 and refers to the value of the residual of the Poisson equation.
Both convergence criteria are set so that the drain current of the device does not vary by more than ~1% (in the worst case) if more self-consistent iterations were allowed.
Mathieu
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George Pau @ on
Thanks for the reply. I have two related questions.
First, which norm are you using? Is it the vector (L2) norm or the max (L_\infty) norm? Based on your last paragraph, I guess ultimately the convergence criterion is based on the computed drain current?
Second, I was reading your paper PRB 74, 205323 (2006). You mentioned in your first paragraph that the carrier density is updated using simple analytical formula. Which formula in the article are you referring to?
Thanks, George
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Mathieu Luisier @ on
1) neither L2 nor L_\infty, but sum_ |Vnew(i)-Vold(i)|, where Vnew is the electrostatic potential at the new iteration and Vold the one at the previous iteration. The index i runs over all the discretization points. The drain current is not used as convergence criterion. It was just verified for different devices that the drain current has converged when the convergence criterion mentioned above is reached.
2) in the PRB 74 paper there is no self-consistent calculation so that no simple analytical formula is described there. In OMEN, the following equation is used given the carrier density n(x,y,z) calculated from the Schrodinger equation:
n(x,y,z) = NC * log(1+exp(e*(Ef(x,y,z)-V(x,y,z))/(kB*Temp)))
where NC is a constant, Ef(x,y,z) a position-dependent pseudo Fermi level, and V(x,y,z) the electrostatic potential
a) first we calculate Ef(x,y,z) as function of the input n(x,y,z). This can be done analytically. b) after we got Vnew(x,y,z), we can easily calculated nnew(x,y,z) using Ef(x,y,z) and Vnew(x,y,z)
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George Pau @ on
Thanks for the answer. But, I find the convergence criterion used to be rather strange. If I increase the resolution of the simulation, the computational time will increase since we are solving a larger problem. This is expected. However, the increase of computational time can also be attributed to a more stringent convergence criterion since we have to sum over more discretization points. Are you sure you don’t divide the sum by the number of discretization points?
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