SEMICONDUCTOR DEVICE PHYSICS AND DESIGN

546 APPENDIX E. BEYOND THE DEPLETION APPROXIMATION

This is called the Gummel correction to the built-in voltage. To apply the depletion approxima-
tion and calculate parameters related to electrostatics such as depletion region width, depletion
capacitance etc., it is necessary to substituteV

′

biforVbiin previous formulae. Hence,

W=

√

2 s

e

(

1

NA

+

1

ND

)

(Vbi′) (E.3)

note that in a Schottky barrier the correction due to the thermal broadening of carriers (which
occurs over a Debye Length,LD) occurs in only the semiconductor and hence

V

′

bi=Vbi−

kBT

e

for a Schottky barrier.
The Gummel correction is arrived at by solving Poisson’s equation in the depletion region in-
cluding the contribution of mobile charges. Consider the band diagram of ap-njunction in
figure E.3. For the purpose of our analysis we will only consider thep-type semiconductor. The
analysis is equivalent for then-side. The governing equations are

d^2 Ψ

dx^2

=−

ρ(x)



(P oisson′sequation) (E.4)

and
ρ(x)=q(ND+−NA−+pp−np) (E.5)

whereND+andNA−are the ionized donors and acceptors respectively with the latter dominant
in thep-region. In the bulk of the semiconductor charge neutrality requiresρ(x)=0or from
equation E.5
ND+−NA−=np 0 −pp 0 (E.6)

Applying equation E.6 to equation E.5 and equation E.4 we get the resultant Poisson’s equation

d^2 Ψ

dx^2

=−

e



[(pp−pp 0 )−(np−np 0 )] (E.7)

From Boltzmann statistics and figure E.3 we knowpp=pp 0 e−
keΨ
BTandnp=np 0 e+
keΨ
BTor

d^2 Ψ

dx^2

=−

e



[

pp 0 (e−

keΨ

BT−1)−np 0 (e+

keΨ

BT−1)

]

(E.8)

Recognizing that

(∂Ψ

∂x

)

d

(∂Ψ

∂x

)

=∂

(^2) Ψ
∂x^2 dΨwe can integrate equation E.8 from the bulk towards
the junction
∫∂∂xΨ
0

(

∂Ψ

∂x

)

d

(

∂Ψ

∂x

)

=−

e



∫Ψ

0

[

pp 0 (e−

keΨ

BT−1)−np 0 (e+

keΨ

BT−1)

]

dΨ (E.9)

UsingE=−∂x∂Ψwe get

E=

(

2 kBT



pp 0

[(

e−

keΨ

BT+ eΨ

kBT

− 1

)

+

np 0

pp 0

(

e−

keΨ

BT− eΨ

kBT

− 1

)])^12

(E.10)