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

$$\rho_{L}^{} {\frac{\partial T_{L}}{\partial t}}$$ div S_L = (3.15)

$$\kappa_{L}^{}$$ S_L = (3.16)

with S_L being the lattice heat flow density. The coefficients of this equation are $\rho_{L}^{}$, c_L, 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,

$$\left(\vphantom{\frac{E_{C}}{\mathrm{q}}-\psi}\right. {\frac{E_{C}}{\mathrm{q}}} \psi \left.\vphantom{\frac{E_{C}}{\mathrm{q}}-\psi}\right) \left(\vphantom{\frac{E_{V}}{\mathrm{q}}-\psi}\right. {\frac{E_{V}}{\mathrm{q}}} \psi \left.\vphantom{\frac{E_{V}}{\mathrm{q}}-\psi}\right)$$ (3.17)

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

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