Page 313 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

Interacting Subsystems
bulk the sum of majority and minority currents of one type add up to
the total current so that in general
x
J = constant = J () + J () (7.168a)
x
N P
In the depletion region, since we will assume no carrier recombina-
tion here, the minority carrier densities add up to give the total current
so that
(
J = constant = J ( x ) + J – x ) (7.168b)
p
P
n
N
• We obtain closure of the relations by obtaining the carrier concentra-
tions and their slopes at the edges of the depletion region. These four
values become boundary conditions for the above expressions. We
obtain an expression for the electric field by considering the drift-dif-
fusion equation in the depletion region together with the Einstein
relation

dn
J ≈ 0 = qµ nE + qD ------ (7.169a)
N n N
dx
D N 1 dn  k T 1 dn 
 
B  
E = – ------- --- ------ = – --------- --- ------ (7.169b)
µ  dx q  dx
n 
n 
n
Since the voltage drop over the depletion region is simply minus the
integral of the electric field, we obtain
x n
d
V depletion = V – V Applied = – ∫ E x
bi
– x p
(7.170)
x n
k T  dn  k T x n k T n x( n )
B
B
n
B
q ∫
= ---------  ------ = ---------ln n () – = ---------ln -------------------
n 
q
q
n –
x p
p
p ( x )
– x p
In thermal equilibrium, where V Applied = 0 , we have from (7.158)
that
2
qV bi n i
exp – ----------- = ---------------- (7.171)
k T n p
B n0 p0
which can be combined with (7.170) to give
310 Semiconductors for Micro and Nanosystem Technology