3.1.4 Lattice Heat Flow Equation

To account for self-heating effects in semiconductor devices, the lattice heat flow equation has to be solved.

$$\mathrm{div}(\kappa_L\cdot\mathrm{grad}\,T_\mathrm{L}) = \rho_{\mathrm{L}}\cdot c_{\mathrm{L}}\cdot \frac{\partial T_\mathrm{L}}{\partial t} - H$$ (3.25)

The coefficients $\rho_{\mathrm{L}}$, $c_{\mathrm{L}}$, and $\kappa_{\mathrm{L}}$ are the mass density, specific heat, and thermal conductivity of the respective materials.

The model for the heat generation, $H$, depends on the transport model employed. In the drift-diffusion case $H$ equals the JOULE heat,

$$H= \mathrm{grad}\left(\frac{\ensuremath{E_\mathrm{c}}}{{\mathrm{q... ...si\right)\cdot{\mathbf{J}}_p + R\cdot(\ensuremath{E_\mathrm{c}}- E_\mathrm{v}),$$ (3.26)

whereas in the energy transport case the relaxation terms are used

$$H = \frac{3\cdot {\mathrm{k_B}}}{2}\cdot\left( n\cdot\frac{T_n -T... ...{\varepsilon,n}} + p\cdot\frac{T_p -T_\mathrm{L}}{\tau_{\varepsilon,p}}\right).$$ (3.27)

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