Electrostatic potential

Poisson Equation

This equation governs the relation between the electrostatic potential $\varphi (\mathbf{x})$ and total charge density distribution $\rho (\mathbf{x},\varphi )$ as follows:

(2.2.1)$$-\mathrm{\nabla }\cdot [{\epsilon }_0{\epsilon }_r(\mathbf{x})\mathrm{\nabla }\cdot \varphi (\mathbf{x})]=\rho (\mathbf{x},\varphi )$$

where ${\epsilon }_0$ is the vacuum permittivity, ${\epsilon }_r$ is the material dependent static dielectric constant. And the total charge density distribution consists of the densities of ionized donors $N_D^{+}$, ionized acceptors $N_D^{-}$, piezoelectric and pyroelectric charge ${\rho }_{pz}$ and ${\rho }_{py}$, besides the carrier densities $n(\mathbf{x},\varphi )$ and $p(\mathbf{x},\varphi )$, which are calculated either classically or quantum mechanically:

(2.2.2)$$\rho (\mathbf{x},\varphi )=e[-n(\mathbf{x},\varphi )+p(\mathbf{x},\varphi )+N_D^{+}(x)-N_A^{-}(\mathbf{x})+{\rho }_{\text{pz}}(\mathbf{x})+{\rho }_{\text{py}}(\mathbf{x})]$$

When the Schrödinger-Poisson equation is solved, i.e. quantum_poisson{ } is specified in run{ } section, the carrier densities defined in either multi-band model or single-band model are substituted into this $\rho (\mathbf{x},\varphi )$ and the Poisson equation is solved accordingly. Then the resulting $\varphi (\mathbf{x})$ is returned into the Schrödinger equation and the carrier densities are calculated once again.

This cycle is continued until the carrier densities satisfies the convergence criteria, which can be tuned by the users from run{ poisson{ } }. The final result of $n(\mathbf{x},\varphi )$, $p(\mathbf{x},\varphi )$ and $\varphi (\mathbf{x})$ must satisfy both Schrödinger and Poisson equations, or we can say that the Schrödinger equation and Poisson equation are self-consistent with respect to the resulting carrier densities and electrostatic potential.

On the other hand, when only the Poisson equation is solved, i.e. only poisson{ } is specified run{ } section, the carrier densities are calculated according to (2.2.3) and (2.2.4) instead. We can say in other words that the carrier density calculation in the context of Thomas-Fermi approximation and the Poisson equation are self-consistent with respect to the resulting carrier densities and electrostatic potential.

Last update: 04/12/2024