SEMICONDUCTOR DEVICE PHYSICS AND DESIGN

174 CHAPTER 4. JUNCTIONS IN SEMICONDUCTORS:P-NDIODES

or
Gth=αrn^2 i

In non equilibrium recognizing thatGthis a function of temperature and invariant under injection

Gth−R=αrn^2 i−αrnp=

pn 0 −pn

τp

(4.5.3)

Therefore, from equation 4.5.1

Dp

d^2 pn

dx^2

+

pn 0 −pn

τp

=0 (4.5.4)

Note that the second term in this sum is a generation term becausepn<pn 0 for allx>Wn.
This is natural because both generation and recombination are mechanisms by which the system
returns to its equilibrium value. When the minority carrier concentration is above the equilibrium
minority carrier value then recombination dominates and when the minority carrier concentration
is less than that at equilibrium, then generation dominates. Again, usingΔpn(x−Wn)=
pn 0 −pn(x−Wn),wehave

Ln X=0 Lp

p n

+

+

Figure 4.17: Minority carriers generated within a diffusion length of the depletion region enter
and are swept away.

Dp

d^2 Δpn(x)

dx^2

+

Δpn(x)

τp

=0 (4.5.5)

and,

Δpn(x−Wn)=C 1 exp

(

+

x−Wn

Lp

)

+C 2 exp

(

−

x−Wn

Lp

)

(4.5.6)

Where we know that:
⎧
⎪⎨

⎪⎩

C 1 =0 for physical reasons,

Δpn(x−Wn)→ 0 x−Wn=∞,

Δpn(x−Wn)→pn 0 −pn(0) =pn 0 x−Wn=0.