The drift diffusion equations

2. Charge Carrier Transport

Charge transport in semiconductors is described by the coupled drift–diffusion equations for electrons and holes. These equations account for carrier motion driven by the electric field, by concentration gradients, and by temperature gradients (thermoelectric effects).

For electrons, the current density is given by:

\[ \boldsymbol{J_n} = q \mu_e n_f \nabla E_c + q D_n \nabla n_f + q \mu_e n_f \frac{\nabla T}{T}, \]

and for holes:

\[ \boldsymbol{J_p} = q \mu_h p_f \nabla E_v - q D_p \nabla p_f - q \mu_h p_f \frac{\nabla T}{T}. \]

Here, \(q\) is the elementary charge, \(\mu_e\) and \(\mu_h\) are the carrier mobilities, and \(D_n\) and \(D_p\) are the diffusion coefficients for electrons and holes. The quantities \(E_c\) and \(E_v\) denote the local conduction and valence band edge energies, which vary spatially due to electrostatic potential variations. The final term in each expression captures thermally driven transport, reflecting the fact that carriers tend to diffuse along temperature gradients.

Charge conservation is enforced by the carrier continuity equations. For electrons:

\[ \nabla \cdot \boldsymbol{J_n} = q \left( R - G + \frac{\partial n}{\partial t} \right), \]

and for holes:

\[ \nabla \cdot \boldsymbol{J_p} = -q \left( R - G + \frac{\partial p}{\partial t} \right). \]

In these equations, \(R\) and \(G\) represent the recombination and generation rates per unit volume, respectively, and the time derivatives capture transient phenomena such as charge build-up or decay. Taken together, the drift–diffusion equations and continuity equations form the foundation of semiconductor device modeling. They describe how carriers respond to electric fields, spatial inhomogeneities in concentration and temperature, and how they are generated and lost through optical absorption, recombination, or trapping processes.

Within OghmaNano, these equations can be solved in 1D, 2D, or 3D depending on the chosen device geometry, and can be coupled directly with Poisson’s equation and recombination models to simulate the full dynamic behavior of advanced semiconductor devices.