3.1.3 Hydrodynamic Transport Equations

In the hydrodynamic transport equations, carrier temperatures are allowed to be different from the lattice temperature [106]. The basic equations (3.10) through (3.12) are augmented by energy balance equations which determine the carrier temperatures. The current relations take the form

$${\mathbf{J}}_n = {\mathrm{q}}\cdot\mu_n\cdot n\cdot\left( \mathrm... ...{N_{C,0}}{n}\cdot\mathrm{grad}\left( \frac{n\cdot T_n}{N_{C,0}}\right) \right),$$ (3.17)

$${\mathbf{J}}_p = {\mathrm{q}}\cdot\mu_p\cdot p\cdot\left( \mathrm... ...{N_{V,0}}{p}\cdot\mathrm{grad}\left( \frac{p\cdot T_p}{N_{V,0}}\right) \right).$$ (3.18)

The energy balance equations state conservation of the average carrier energies. In terms of the carrier temperatures, $T_n$ and $T_p$, they can be written as

$$\mathrm{div}\ {\mathbf{S}}_n = \mathrm{grad}\left(\frac{\ensurema... ...ial t} + R\cdot T_n + n\cdot\frac{T_n -T_\mathrm{L}}{\tau_{\epsilon,n}}\right),$$ (3.19)

$$\mathrm{div}\ {\mathbf{S}}_p = \mathrm{grad}\left(\frac{E_\mathrm... ...tial t} + R\cdot T_p +p\cdot\frac{T_p -T_\mathrm{L}}{\tau_{\epsilon,p}}\right).$$ (3.20)

Here, $\tau_{\epsilon,n}$ and $\tau_{\epsilon,p}$ denote the energy relaxation times, while ${\mathbf{S}}_n$ and ${\mathbf{S}}_p$ are the energy fluxes.

$${\mathbf{S}}_n = -\kappa_n\cdot\mathrm{grad}\ T_n - \frac{5}{2}\cdot\frac{{\mathrm{k_B}}\cdot T_n} {{\mathrm{q}}}\cdot{\mathbf{J}}_n\,,$$ (3.21)

$${\mathbf{S}}_p = -\kappa_p\cdot\mathrm{grad}\ T_p + \frac{5}{2}\cdot\frac{{\mathrm{k_B}}\cdot T_p}{ {\mathrm{q}}}\cdot{\mathbf{J}}_p\,.$$ (3.22)

The thermal conductivities, $\kappa_n$ and $\kappa_p$, are assumed to obey a generalized WIEDEMANN-FRANZ law [107].

$$\kappa_n = \left(\frac{5}{2} + c_n\right)\cdot\frac{{\mathrm{k_B}}^2}{{\mathrm{q}}} \cdot T_n\cdot \mu_n\cdot n\,,$$ (3.23)

$$\kappa_p = \left(\frac{5}{2} + c_p\right)\cdot\frac{{\mathrm{k_B}}^2}{{\mathrm{q}}} \cdot T_p\cdot \mu_p\cdot p\,,$$ (3.24)

where $c_n$ and $c_p$are electron and hole characteristic exponents of the relaxation time dependence on the scattering mechanisms, respectively. Strictly speaking, this model represents an energy transport model. Such a model is obtained when in the course of deriving the moment equations the average kinetic energy is consequently neglected against the thermal energy, assuming that $\frac{1}{2}\cdot m_{\nu}\cdot v_{\nu}^{2}\ll{\mathrm{k_B}}\cdot T_{\nu}$. T. Ayalew: SiC Semiconductor Devices Technology, Modeling, and Simulation