SEMICONDUCTOR DEVICE PHYSICS AND DESIGN

APPENDIX E. BEYOND THE DEPLETION APPROXIMATION 547

For our purposes of understanding the origin of the Gummel correction we will evaluate equation
E.10 for a condition of mild depletion, whereΨis small and positive, of such magnitude that

E=

(

2 kBT



pp 0

) 12 (

eΨ

kBT

− 1

) (^12)
(E.11)
where the 2ndterm in parentheses in equation E.10 is neglected because of thenppp^00 pre-factor
and thee−
keΨ
BTis neglected becauseΨis positive. Thus

E=

√

2 pp 0

e

(

Ψ−

kBT

e

)

This is identical to the depletion approximation except forΨbeing replaced byΨ−kBeT.This
reflects the reduced electric field because of the effect of mobile charges (in our case holes) at
the depletion region edge.
Therefore, the depletion region edge is defined by using the depletion approximation while
reducing the built-in potential bykBqTat each depletion region edge as shown in figure E.2 and
stated in equation E.3.

eΨ

Ecp

Eip

Evp

Ecn

Ein

Evn

E EF

F

Figure E.3: Band diagram of ap-njunction showing the references used to describeeΨ.