3.1.2 The Drift-Diffusion Transport Model

The drift-diffusion current relations can, amongst others, be derived from the Boltzmann transport equation by the method of moments [33] or from basic principles of irreversible thermodynamics [75]. The electron and hole current densities are given by

$$\mathbf{J}_n = \mathrm{q}\cdot\mu_n\cdot n\cdot\left(\mathrm{grad... ...\cdot\mathrm{grad}\left(\frac{n\cdot T_{{\mathrm{L}}}}{N_{C,0}}\right) \right),$$ (3.4)

$$\mathbf{J}_p = \mathrm{q}\cdot\mu_p\cdot p\cdot\left(\mathrm{grad... ...\cdot\mathrm{grad}\left(\frac{p\cdot T_{{\mathrm{L}}}}{N_{V,0}}\right) \right).$$ (3.5)

These current relations account for position-dependent band edge energies, $E_{C}$ and $E_{V}$, and for position-dependent effective masses, which are included in the effective density of states, $N_{C,0}$ and $N_{V,0}$. The index $0$ indicates that $N_{C,0}$ and $N_{V,0}$ are evaluated at some (arbitrary) reference temperature, $\mathrm {T_0}$, which is constant in real space regardless of what the local values of the lattice and carrier temperatures are.