3.1.3 Hydrodynamic Model Current Equations

For the hydrodynamic (HD) transport model used in this work the current densities are obtained for electrons as:

$${\bf {J}}_n = q \cdot \mu_n \cdot n \cdot \bigg( {\text{grad}} \b... ...{N_{C,0}}{n} \cdot {\text{grad}} \bigg( \frac{n\cdot T_n}{N_{C,0}}\bigg )\bigg)$$ (3.6)

and for holes:

$${\bf {J}}_p = q \cdot \mu_p \cdot p \cdot \bigg( {\text{grad}} \b... ...{N_{V,0}}{p} \cdot {\text{grad}} \bigg( \frac{p\cdot T_p}{N_{V,0}}\bigg )\bigg)$$ (3.7)

Comparing to the DD approach the carrier temperature ${\it T}_\mathrm{\nu}$ can deviate from the lattice temperature ${\it T}_\mathrm{L}$. The conservation of the average energy $w_\nu$ is written:

$${\bf {S}}_n = \bigg(\frac{E_C}{q}-\psi \bigg)\cdot {\bf {J}}_n - \bigg(\frac{\p... ...}}}{\tau_{w,n}}+ \frac{3 \cdot {\it k}_{\mathrm{B}}}{2} \cdot R\cdot T_n \bigg)$$ (3.8)

$${\bf {S}}_p = \bigg(\frac{E_V}{q}-\psi \bigg)\cdot {\bf {J}}_p - \bigg(\frac{\p... ...}}}{\tau_{w,p}}+ \frac{3 \cdot {\it k}_{\mathrm{B}}}{2} \cdot R\cdot T_p \bigg)$$ (3.9)

The energy fluxes $S_\nu$ are defined as:

$${\bf {S}}_n=-\kappa_n \cdot T_n - \frac{1}{q} \cdot (w_n + {\it k}_{\mathrm{B}}\cdot T_n)\cdot {\bf {J}}_n$$ (3.10)

$${\bf {S}}_p=-\kappa_p \cdot T_p + \frac{1}{q} \cdot (w_p + {\it k}_{\mathrm{B}}\cdot T_p) \cdot {\bf {J}}_p$$ (3.11)

$w_n$ and $w_p$ denote the average carrier energies, $w_{\nu,0}$ the equilibrium energies. The $w_\nu$ are approximated as:

$$w_\nu = \frac{3}{2}\cdot {\it k}_{\mathrm{B}}\cdot T_\nu + \frac{... ...ot m_\nu \cdot v_\nu^2 \approx \frac{3}{2}\cdot {\it k}_{\mathrm{B}}\cdot T_\nu$$ (3.12)

The first term represents the carrier thermal energy while the second denotes the carrier kinetic energy. In this work the carrier kinetic energy is neglected against the thermal energy. Some publications name the simplified transport model an energy transport model. The impact of this simplification, which goes along with a limited number of moments considered for the solution of the Boltzmann equation, and the results for device application from both numerical and accuracy aspects are discussed in Section 3.6. The carrier thermal conductivities $\kappa_\nu$ are defined for electrons:

$$\kappa_n  = \bigg(\frac{5}{2}+c_n\bigg) \cdot \frac{{\it k}_{\mathrm{B}}^2}{q}\cdot T_n \cdot \mu_n \cdot n$$ (3.13)

and for holes:

$$\kappa_p  = \bigg(\frac{5}{2}+ c_p \bigg) \cdot \frac{{\it k}_{\mathrm{B}}^2}{q}\cdot T_p \cdot \mu_p \cdot p$$ (3.14)

(3.14) and (3.15) represent a generalized Wiedemann-Franz law. The heat capacities $c_n$ and $c_p$ of the electron and hole gases are typically neglected.