3.1.3 The Hydrodynamic Transport Model

In the hydrodynamic transport model, carrier temperatures are allowed to be different from the lattice temperature. The basic equations (3.1) through (3.3) are augmented by energy balance equations which determine the carrier temperatures. The current relations take the form

$$\mathbf{J}_n = \mathrm{q}\cdot\mu_n\cdot n\cdot\left(\mathrm{grad... ...c{N_{C,0}}{n}\cdot\mathrm{grad}\left(\frac{n\cdot T_n}{N_{C,0}}\right) \right),$$ (3.6)

$$\mathbf{J}_p = \mathrm{q}\cdot\mu_p\cdot p\cdot\left(\mathrm{grad... ...c{N_{V,0}}{p}\cdot\mathrm{grad}\left(\frac{p\cdot T_p}{N_{V,0}}\right) \right).$$ (3.7)

The energy balance equations state conservation of the average carrier energies. In terms of the carrier temperatures $T_n$ and $T_p$ they can be written as

$$\mathrm{div} \mathbf{S}_n = \mathrm{grad}\left(\frac{E_{C}}{\mat... ...t} + R\cdot T_n + n\cdot\frac{T_n -T_{{\mathrm{L}}}}{\tau_{\epsilon,n}}\right),$$ (3.8)

$$\mathrm{div} \mathbf{S}_p = \mathrm{grad}\left(\frac{E_{V}}{\mat... ...t} + R\cdot T_p + p\cdot\frac{T_p -T_{{\mathrm{L}}}}{\tau_{\epsilon,p}}\right).$$ (3.9)

Here, $\tau_{\epsilon,n}$ and $\tau_{\epsilon,p}$ denote the energy relaxation times, while $\mathbf{S}_n$ and $\mathbf{S}_p$ are the energy fluxes.

$$\mathbf{S}_n = -\kappa_n\cdot\mathrm{grad} T_n - \frac{5}{2}\cdot\frac{\mathrm{k_B}\cdot T_n} {\mathrm{q}}\cdot\mathbf{J}_n$$ (3.10)

$$\mathbf{S}_p = -\kappa_p\cdot\mathrm{grad} T_p + \frac{5}{2}\cdot\frac{\mathrm{k_B}\cdot T_p} {\mathrm{q}}\cdot\mathbf{J}_p$$ (3.11)

The carrier thermal conductivities, $\kappa_n$ and $\kappa_p$, are assumed to obey a generalized Wiedemann-Franz law.

$$\kappa_n = \left(\frac{5}{2} + c_n\right)\cdot\frac{\mathrm{k_B}^2}{\mathrm{q}} \cdot T_n\cdot \mu_n\cdot n$$ (3.12)

$$\kappa_p = \left(\frac{5}{2} + c_p\right)\cdot\frac{\mathrm{k_B}^2}{\mathrm{q}} \cdot T_p\cdot \mu_p\cdot p$$ (3.13)

The heat capacities $c_n$ and $c_p$ are usually neglected ($c_n=c_p=0$).

Strictly speaking, the model described above in this subsection represents an energy transport (ET) model. Such a model is obtained when in the course of deriving the moment equations the average kinetic energy is consequently neglected against the thermal energy, assuming that $\frac{1}{2}\cdot m_{\nu}\cdot v_{\nu}^{2} \ll \mathrm{k_B}\cdot T_{\nu}$. Since in the literature the terms energy transport and hydrodynamic transport are often used synonymously, and since during the development of MINIMOS-NT the term hydrodynamic transport was used, this notation is retained throughout this work.