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

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

whereas in the hydrodynamic case the relaxation terms are used
    $\displaystyle H = \frac{3\cdot \mathrm{k_B}}{2}\cdot\left(n\cdot\frac{T_n -T_{{...
...t\frac{T_p -T_{{\mathrm{L}}}}{\tau_{\epsilon,p}}\right) + R\cdot(E_{C}- E_{V}).$ (3.17)


next up previous contents
Next: 3.1.5 The Insulator Equations Up: 3.1 Sets of Partial Previous: 3.1.3 The Hydrodynamic Transport
Vassil Palankovski
2001-02-28