2.2 Hydrodynamic Transport Model

In the isotropic effective-mass approximation, the left-hand side of (2.1) can be used to deduce equations for electron density, drift-velocity and mean energy as defined in (2.3),(2.4), and (2.5) corresponding to equations of moments of zero, first and second-order in the wave vector. In order to close the hierarchy of equations a Maxwellian with electron temperature T_n is assumed for the distribution function

(2.6)

Here, the prefactor A determines the carrier density, while the carrier temperature T_n determines the shape of the distribution function. Approximating the effect of scattering through the macroscopic relaxation time approximation (with relaxation time τ_v for the drift velocity and τ_w for the mean energy) results in (e.g. [153])

$\displaystyle =$	$\displaystyle 0,$	(2.7)
$\displaystyle =$		(2.8)
$\displaystyle =$		(2.9)

Here, the mean energy is related to the electron T_n and the lattice temperature T_L by

$\displaystyle w$	$\displaystyle =$		(2.10)
$\displaystyle w_0$	$\displaystyle =$		(2.11)

respectively. The drift energy m^*v²/2 is typically small compared to the thermal energy 3k_BT_n/2. As a third-order moment the heat flux Q cancels for a Maxwellian, but is often assumed phenomenologically to be proportional to the gradient of the electron temperature. In the stationary regime the drift energy is negligible. After introducing the conduction current density and , one obtains

	$\displaystyle =$	$\displaystyle 0,$	(2.12)
	$\displaystyle =$		(2.13)
			(2.14)
$\displaystyle \mathbf{S}$			(2.15)

with the electron mobility . These system along with Poisson's equation (2.2) form the well known HD model.

I. Starkov: Comprehensive Physical Modeling of Hot-Carrier Induced Degradation