Page 229 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

Transport Theory
where we scaled the electrostatic potential in terms of the thermal voltage
according to
qψ
u = --------- (6.93)
k T
B
The r.h.s. of (6.92) is a product of the nodal value ρ of the charge den-
k
k
sity and the area of the portion of the box surrounding node that lies in
the element. The discretization of (6.90b) and (6.90c) has to be per-
formed with care since we have already made assumptions on the poten-
tial. Introducing (6.89a) and (6.93) into (6.88a) yields the electron
current density

(
–
j = qD ∇nn∇u) (6.94)
n n
j may be projected on an edge of the triangular element, say L , to
n k
give
 ∂n du 
j = qD – n = j l (6.95)
nk n l ∂  k l d  n k
k
The projection j nk is assumed to be constant along the edge L k . This
constrains the element size to a length scale where the assumption is ful-
filled. On the other hand, the linear potential function on the edge

u – u i
j
ul() = ---------------l + u = a l + u (6.96)
k k i k k i
L k
leads to a differential equation for along L k
n
dn j nk
J nk = – α n = ---------- (6.97)
k
l d qD
k n
Integrating (6.97) from node i to node j yields
(
(
⁄
n
J = [ Bu )n – Bu )n ] L . n and n are the values of at nodes
nk ji j ij i k i j
⁄
–
j
i and , respectively. Bu( ) = x ( exp u ( ) 1) is the Bernoulli func-
ji ji
tion for the argument u = u – u . Thus the flux of j related to node
ji j i n
k enters (6.90b) to give
226 Semiconductors for Micro and Nanosystem Technology