Page 223 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

Transport Theory
We have encountered several of the levels and descriptions up to now.
The only job left is to describe the global balance picture.
6.3.1 Global Balance Equation Systems
In a global balance equation system we consider only the contacts and
integrate over their cross-sections to obtain currents instead of current
densities. At this level we have lost the distributed nature of transport
phenomena. Global balance simulation programs treat networks of
devices that are connected through one–dimensional idealized wires.
These wires do not dissipate energy and all functionality is concentrated
in the single devices. The resulting equation of motion is no longer a par-
tial differential equation but rather a system of ordinary differential equa-
tions. This is why we call such systems concentrated parameter systems
or lumped models.

6.3.2 Local Balance: The Hydrodynamic Equations

Recall the balance equations for the moments (6.19), 6.21) and 6.21), and
rewrite them in a slightly different manner, i.e., put j = nv , u = nw ,
n a
and j = nwv . Here we introduced w as the average energy of the
u a
electrons and used the average velocity v as given in (6.11). The colli-
a
sion terms on the r.h.s. of the balance equations are interpreted as rates of
changes due to collisions for the carrier density, the current density and
the energy, respectively. Therefore, we write

 ∂n 
C = t ∂   C (6.77a)
n

 ∂ nv d   ∂n   ∂v d 
C = C = ----------- = v + n (6.77b)
j n p  t ∂  C d t ∂   C t ∂   C
 ∂ nw   ∂n   ∂w 
C = C w =  ---------- = w t ∂   + n t ∂   (6.77c)
t ∂ 
u
C C C
Thus we have for the density balance equation

220 Semiconductors for Micro and Nanosystem Technology