Solving Poisson's Equation for MOSFETs: Analytic vs. Finite Difference Approach

[Excom]

Hi

I am working with MOSFETs and in this context I am trying to solve poissons equation inside a MOSFET. Only in the direction from the gate through the oxide and into the silicon. I know the analytic solution but now I want solve Poissons equation with the use of finite difference.

When I found the analytic solution, I utilized that the potential and the derivative of the potential is zero deep into the body.

When I make the finite difference scheme, how do I ensure that the potential and the derivative of the potential is zero deep into the silicon?

Thanks in advance

 

[eljose79]

Dear fellow scientist,

Thank you for your question regarding solving Poisson's equation inside a MOSFET using finite difference. This is a common challenge in the field of semiconductor device simulation and I am happy to offer some guidance on this topic.

Firstly, it is important to understand that the boundary conditions for Poisson's equation in this context will depend on the device structure and the specific physical effects that are being considered. Therefore, there is no one-size-fits-all solution for ensuring that the potential and its derivative are zero deep into the silicon.

One approach is to use a Dirichlet boundary condition at the silicon-oxide interface, where the potential is set to a constant value based on the voltage applied to the gate. This can be done by imposing a finite difference stencil at the interface that takes into account the oxide thickness and material properties. This will ensure that the potential is constant at the interface and will gradually decrease towards zero as you move deeper into the silicon.

Another approach is to use a Neumann boundary condition, where the derivative of the potential is set to zero at the interface. This can be achieved by using a finite difference stencil that takes into account the electric field at the interface, which is related to the voltage applied to the gate and the oxide thickness. This will ensure that the electric field is zero at the interface and will gradually increase as you move deeper into the silicon.

Ultimately, the best approach will depend on the specific device structure and physical effects you are trying to simulate. I recommend consulting with colleagues or references in your field to determine the most appropriate boundary conditions for your particular case.

I hope this helps and wish you success in your research.

 

[charng]

.

Dear researcher,

Thank you for sharing your work on solving Poisson's equation for MOSFETs. It is a crucial step in understanding and optimizing the performance of these devices.

In terms of your question about ensuring that the potential and its derivative are zero deep into the silicon when using a finite difference approach, there are a few methods you can consider.

Firstly, you can define the boundary conditions at the deep end of the silicon as zero potential and zero derivative, and then use these as starting points for your finite difference calculations. This will ensure that your solution remains consistent with the known behavior at that boundary.

Another approach is to use a large number of grid points in your finite difference scheme, which will allow for a more accurate representation of the potential and its derivative at the deep end of the silicon. This can help to mitigate any errors that may arise from using a smaller number of grid points.

Finally, you can also use a relaxation method, such as the Gauss-Seidel method, to iteratively solve for the potential and its derivative at each grid point until they converge to the desired values. This can be a more computationally intensive approach, but it can also provide more accurate results.

I hope these suggestions are helpful in addressing your question. Good luck with your research!