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 or from basic principles of irreversible thermodynamics under the assumption T_$\nu$ = T_L. The electron and hole current densities are given by

$$\mu_{n}^{} \left(\vphantom{\mathrm{grad}\left(\frac{E_{C}}{\mathrm{q}}-\psi\right) + \fra... ...N_{C,0}}{n}\cdot\mathrm{grad}\left(\frac{n\cdot T_{L}}{N_{C,0}}\right) }\right. \left(\vphantom{\frac{E_{C}}{\mathrm{q}}-\psi}\right. {\frac{E_{C}}{\mathrm{q}}} \psi \left.\vphantom{\frac{E_{C}}{\mathrm{q}}-\psi}\right) {\frac{\mathrm{k_{B}}}{\mathrm{q}}} {\frac{N_{C,0}}{n}} \left(\vphantom{\frac{n\cdot T_{L}}{N_{C,0}}}\right. {\frac{n\cdot T_{L}}{N_{C,0}}} \left.\vphantom{\frac{n\cdot T_{L}}{N_{C,0}}}\right) \left.\vphantom{\mathrm{grad}\left(\frac{E_{C}}{\mathrm{q}}-\psi\right) + \fra... ...N_{C,0}}{n}\cdot\mathrm{grad}\left(\frac{n\cdot T_{L}}{N_{C,0}}\right) }\right)$$ (3.5)

$$\mu_{p}^{} \left(\vphantom{\mathrm{grad}\left(\frac{E_{V}}{\mathrm{q}}-\psi\right) - \fra... ...N_{V,0}}{p}\cdot\mathrm{grad}\left(\frac{p\cdot T_{L}}{N_{V,0}}\right) }\right. \left(\vphantom{\frac{E_{V}}{\mathrm{q}}-\psi}\right. {\frac{E_{V}}{\mathrm{q}}} \psi \left.\vphantom{\frac{E_{V}}{\mathrm{q}}-\psi}\right) {\frac{\mathrm{k_{B}}}{\mathrm{q}}} {\frac{N_{V,0}}{p}} \left(\vphantom{\frac{p\cdot T_{L}}{N_{V,0}}}\right. {\frac{p\cdot T_{L}}{N_{V,0}}} \left.\vphantom{\frac{p\cdot T_{L}}{N_{V,0}}}\right) \left.\vphantom{\mathrm{grad}\left(\frac{E_{V}}{\mathrm{q}}-\psi\right) - \fra... ...N_{V,0}}{p}\cdot\mathrm{grad}\left(\frac{p\cdot T_{L}}{N_{V,0}}\right) }\right)$$ (3.6)

Here, $\mu_{n}^{}$ and $\mu_{p}^{}$ denote the carrier mobilities, and T_L is the lattice temperature. These current relations account for position-dependent band edge energies, E_C and E_V, and 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, T₀, which is constant in real space regardless of what the local values of the lattice and carrier temperatures are.