(7.17) 
Selfconsistency in electronic device simulation means ensuring that the carrier density profile in the simulated device is consistent with its potential profile. The selfconsistent field effectively acts to screen the positive applied voltage. Therefore, the effects of the selfconsistent field can essentially be modeled by a smaller applied bias.
The Poisson and the von Neumann equation form a system of coupled nonlinear differential equations which is usually solved by an iteration between Poisson's and von Neumann's equation.
The potential which enters the Schrödinger equation (see Figure 7.1) is the sum of the solution of the Poisson equation and an from the materials' bandgap
(7.18) 
(7.19) 

In order to solve the coupled system, we have used the Gummel method [Gum64], which takes into account the exponential dependence of the density on the potential. We follow the approach from [BAPGR02].
With this the step for updating the potential in the iteration is given by
(7.20) 
Further details can be found in [BAPGR02], where this method, which is based on classical assumptions, was introduced.
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