next up previous contents
Next: 3.1.5 The Constant Quasi-Fermi Up: 3.1 Sets of Partial Previous: 3.1.3 The Hydrodynamic Transport

3.1.4 The Lattice Heat Flow Equation

To account for self-heating effects in semiconductor devices, the lattice heat flow equation has to be solved which reads
div SL = H - $\displaystyle \rho_{L}^{}$ . cL . $\displaystyle {\frac{\partial T_{L}}{\partial t}}$ (3.15)
SL = - $\displaystyle \kappa_{L}^{}$ . grad TL (3.16)

with SL being the lattice heat flow density. The coefficients of this equation are $ \rho_{L}^{}$, cL, and $ \kappa_{L}^{}$, which denote the materials mass density, specific heat, and thermal conductivity, respectively. H is the generated local heat density and is modeled in dependence of the transport model. In the drift-diffusion case H equals the Joule heat,

H = grad$ \left(\vphantom{\frac{E_{C}}{\mathrm{q}}-\psi}\right.$$ {\frac{E_{C}}{\mathrm{q}}}$ - $ \psi$ $ \left.\vphantom{\frac{E_{C}}{\mathrm{q}}-\psi}\right)$ . Jn + grad$ \left(\vphantom{\frac{E_{V}}{\mathrm{q}}-\psi}\right.$$ {\frac{E_{V}}{\mathrm{q}}}$ - $ \psi$ $ \left.\vphantom{\frac{E_{V}}{\mathrm{q}}-\psi}\right)$ . Jp   , (3.17)

whereas in the hydrodynamic case the relaxation terms are used [36]

H = $ {\frac{3\cdot \mathrm{k_{B}}}{2}}$ . $ \left(\vphantom{n\cdot\frac{T_{n} -T_{L}}{\tau_{\epsilon,n}} + p\cdot\frac{T_{p} -T_{L}}{\tau_{\epsilon,p}}}\right.$n . $ {\frac{T_{n} -T_{L}}{\tau_{\epsilon,n}}}$ + p . $ {\frac{T_{p} -T_{L}}{\tau_{\epsilon,p}}}$ $ \left.\vphantom{n\cdot\frac{T_{n} -T_{L}}{\tau_{\epsilon,n}} + p\cdot\frac{T_{p} -T_{L}}{\tau_{\epsilon,p}}}\right)$   . (3.18)


next up previous contents
Next: 3.1.5 The Constant Quasi-Fermi Up: 3.1 Sets of Partial Previous: 3.1.3 The Hydrodynamic Transport
Tibor Grasser
1999-05-31