The drift diffusion equations

2. Electrostatics

In semiconductor device modeling, the local energy levels of the conduction and valence bands (or, equivalently, the LUMO and HOMO in organic semiconductors) are shifted by the electrostatic potential \(\phi\). These band edges are defined as:

\[E_{\mathrm{LUMO}} = -\chi - q\phi\]

\[E_{\mathrm{HOMO}} = -\chi - E_g - q\phi\]

Here \(\chi\) is the electron affinity, \(E_g\) is the band gap, and \(q\) is the elementary charge. The potential \(\phi\) is determined self-consistently by solving Poisson’s equation across the device.

Poisson’s equation takes the form

\[ \nabla \cdot \bigl( \epsilon_0 \epsilon_r \nabla \phi \bigr) = -q \left( n_f + n_t - p_f - p_t - N_{ad} + N_{ion} + a \right), \]

where \(n_f\) and \(p_f\) are the densities of free electrons and holes, while \(n_t\) and \(p_t\) are the corresponding trapped carrier densities. The term \(N_{ad}\) represents the ionized dopant density, \(N_{ion}\) accounts for background ionic charge (for example, fixed ions in perovskite layers), and \(a\) denotes mobile ions that can drift in response to the local field. The permittivities \(\epsilon_0\) and \(\epsilon_r\) set the strength of the electrostatic response.

This formulation captures all relevant contributions to space charge: free and trapped carriers, intentional doping, and both static and mobile ionic species. It is particularly important for hybrid materials such as organics and halide perovskites, where ionic motion and trap states strongly influence the electrostatic potential and give rise to phenomena such as hysteresis and slow transients.