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Poisson's Equation in Semiconductors

Relates the electrostatic potential (V\text{V}) to the net charge density (ρ\rho) within the semiconductor material: \nabla^2 \text{V} = -\frac{\rho}{\text{\epsilon}_r \text{\epsilon}_0}. This governs built-in potentials.
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The statement of the theorem

The electrostatic potential V(r)V(\mathbf{r}) within a semiconductor material is governed by Poisson's equation, relating the Laplacian of the potential to the net charge density ρ(r)\rho(\mathbf{r}): \nabla^2 V = -\frac{\rho}{\text{\epsilon}_r \text{\epsilon}_0} where ρ=q(p−n+ND−NA)\rho = q(p - n + N_D - N_A) is the net charge density, qq is the elementary charge, and \text{\epsilon}_r is the relative permittivity.
Source: Wikipedia