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 electric fields, concentration gradients, and temperature gradients (thermoelectric effects). A detailed first-principles derivation of these equations from the Boltzmann Transport Equation is given in Drift–Diffusion Theory: From Boltzmann Transport to Energy Balance .

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, \(n_f\) and \(p_f\) are the free electron and hole densities, \(\mu_e\) and \(\mu_h\) are the carrier mobilities, and \(D_n\) and \(D_p\) are the diffusion coefficients. The quantities \(E_c\) and \(E_v\) denote the local conduction- and valence-band edge energies. Writing the current in terms of band-edge gradients rather than the electric field ensures that heterojunctions and material offsets are treated correctly; see Section 5 of the drift–diffusion derivation for details.

The final term in each expression represents thermal driving (thermodiffusion), which arises naturally when the momentum balance is reduced in a self-consistent way. This term is often omitted in simplified models but becomes important in devices with strong temperature gradients or carrier heating. Its origin is discussed in the energy-transport extension .

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:

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

These continuity equations are obtained by taking the zeroth moment of the Boltzmann Transport Equation, as shown explicitly in Section 4 of the derivation . The terms \(R\) and \(G\) represent recombination and generation, while the time derivatives describe transient charge storage and release.

Taken together, the drift–diffusion current relations and continuity equations form the core of semiconductor device modelling. Within OghmaNano, these equations can be solved self-consistently with Poisson’s equation in 1D, 2D, or full 3D, and can be extended to include energy transport (hot-carrier effects) and non-equilibrium trap dynamics when required.