3.1.4 The Lattice Heat Flow Equation

MINIMOS-NT accounts for self-heating effects in semiconductor devices by solving the lattice heat flow equation self-consistently with the DD or HD transport equations, forming together a system of four, or respectively six, partial differential equations.

$$\mathrm{div} \mathbf{S}_{\mathrm{L}}= -\rho_{\mathrm{L}}\cdot c_{\mathrm{L}}\cdot \frac{\partial T_{{\mathrm{L}}}}{\partial t} + H$$ (3.14)

$$\mathbf{S}_{\mathrm{L}}= -\kappa_{\mathrm{L}}\cdot\mathrm{grad} T_{{\mathrm{L}}}$$ (3.15)

In (3.14) $T_{\mathrm{L}}$ denotes the lattice temperature, $t$ is the time variable, and $H$ is the heat generation term. 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 used. In the drift-diffusion case $H$ equals the Joule heat,

$$H = \mathrm{grad}\left(\frac{E_{C}}{\mathrm{q}} -\psi\right)\cdot... ...(\frac{E_{V}}{\mathrm{q}} -\psi\right)\cdot\mathbf{J}_p + R\cdot(E_{C}- E_{V}),$$ (3.16)

whereas in the hydrodynamic case the relaxation terms are used

$$H = \frac{3\cdot \mathrm{k_B}}{2}\cdot\left(n\cdot\frac{T_n -T_{{... ...t\frac{T_p -T_{{\mathrm{L}}}}{\tau_{\epsilon,p}}\right) + R\cdot(E_{C}- E_{V}).$$ (3.17)