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Is the Jacobi matrix in non-linear poisson slover a positivie-definite matrix?

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I have read the papers written by Dr.Zhibin Ren :“NANOSCALE MOSFETS: PHYSICS, SIMULATION AND DESIGN”. and trying to realize the non-linear Poisson solver by myself. when...

http://nanohub.org/answers/question/713

What is the relationship between the potential obtained through the Poisson equation and the vacuum energy level in semiconductor band diagrams?