3.1.2 Drift-Diffusion Current Equations

The popular drift-diffusion model can be derived directly from Boltzmann's transport equation by the method of moments [104] or from the basic principles of irreversible thermodynamics [105]. In this model the electron current density is expressed as a sum of two components: The drift component which is driven by the electric field and the diffusion component caused by the gradient of the electron concentration

$$\mathbf{J} = {\mathrm{q}}\cdot \left( n\cdot\mu\cdot \mathbf{E}+ D_n\cdot \mathrm{grad}\,n \right)$$ (3.13)

where $\mu$ and $D_n$ are the mobility and the diffusivity of the electron gas, respectively. It is clear from the above reasoning that for anisotropic materials $\mu$ and $D_n$ are all tensors of second rank and have the same form as the representative tensor $\sigma$ in (3.2). They are related by the Einstein relation

$$D_n = \mu\cdot\frac{{\mathrm{k_B}}\cdot T_n}{{\mathrm{q}}}$$ (3.14)

where ${\mathrm{k_B}}$ is the Boltzmann constant and $T_n=T_\mathrm{L}$ the lattice temperature which is constant as the electron gas at drift diffusion is assumed to be in thermal equilibrium.

The current relation (3.13) is inserted into the continuity (3.11) and (3.12) to give a second order parabolic differential equation which is then solved together with POISSON's equation (3.10). More generally, according to the phenomenological equations of drift-diffusion the electron and hole current densities ${\mathbf{J}}_n$ and ${\mathbf{J}}_p$ can be expressed as

$${\mathbf{J}}_n = {\mathrm{q}}\cdot\mu_n\cdot n\cdot\left( \mathrm... ...cdot\frac{N_{C,0}}{n}\cdot\mathrm{grad}\left( \frac{n}{N_{C,0}}\right) \right),$$ (3.15)

$${\mathbf{J}}_p = {\mathrm{q}}\cdot\mu_p\cdot p\cdot\left( \mathrm... ...cdot\frac{N_{V,0}}{p}\cdot\mathrm{grad}\left( \frac{p}{N_{V,0}}\right) \right).$$ (3.16)

These current relations account for position-dependent band edge energies, $\ensuremath{E_\mathrm{c}}$ and $E_\mathrm{v}$, 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, $\mathrm{T_0}$, which is constant in real space regardless of what the local values of the lattice and carrier temperatures are.

T. Ayalew: SiC Semiconductor Devices Technology, Modeling, and Simulation