2.1.2 Poisson Equation

By definition the potential $V$ fulfills

$${\mathbf{E}} = -\nabla V .$$ (2.7)

By Maxwell's equations we have

$${\mathbf{D}} = \varepsilon \varepsilon_0 {\mathbf{E}} .$$ (2.8)

and

$$\nabla \cdot {\mathbf{D}}= \rho .$$ (2.9)

Here $\varepsilon_0$ is the electrical permittivity of free space and the dimensionless dielectric constant $\varepsilon$ is a material dependent parameter. In our case the charge density $\rho$ is given by

$$\rho = q(M_0 - C) ,$$ (2.10)

with $q$ the (negative) elementary charge, $M_0$ the zeroth moment, that is, the particle density, and $C$ the net doping.