SEMICONDUCTOR DEVICE PHYSICS AND DESIGN

4.2.P-NJUNCTION IN EQUILIBRIUM 153

Please note that in the above equation, and throughout this chapter, thep−type semiconductor
(the semiconductor on the left hand side) is the reference electrode. In the case of the MOSFET
as we will see in chapter 9 a different reference is used. As noted earlier charge neutrality gives
us
NdWn=NaWp (4.2.21)

andweget

Wp(Vbi)=

{

2 Vbi

e

[

Nd

Na(Na+Nd)

]} 1 / 2

(4.2.22)

Wn(Vbi)=

{

2 Vbi

e

[

Na

Nd(Na+Nd)

]} 1 / 2

(4.2.23)

W(Vbi)=

[

2 Vbi

e

(

Na+Nd

NaNd

)] 1 / 2

(4.2.24)

Theexpressionsderivedabovecanbeextendedtofindtheelectricfields,potential,and
depletionwidthsforarbitraryvaluesofVpandVnundercertainapproximations.Thuswecan
usetheseequationsdirectlywhenthediodeisunderexternalbiasV,bysimplyreplacingVbiby
Vbi-V.
In figure 4.4 we show the charge and electric field profile. The electric field is nonuniform in
the depletion region, peaking at the junction with a peak value.

Em=−

eNdWn



=−

eNaWp



(4.2.25)

The sign of the field simply reflects the fact that in our study the field points toward the
negativex-axis. It is important to note that ifNaNd, the depletion widthWpis much smaller
thanWn. Thus a very strong field exists over a very narrow region in the heavily doped side of
the junction.Insuchjunctions(p+norn+p)thedepletionregionexistsprimarilyonthelightly
dopedside.

Example 4.1A diode is fabricated on ann-type (Nd=10^16 cm−^3 ) silicon substrate, on

which ap-type region doped to 1018 cm−^3 is created. Calculate the Fermi level positions

in thep-andn-regions, determine the contact potential in the diode, and calculate the

depletion widths on thep-andn-side. Using the effective density of states relations, we

have (Nc=2. 8 × 1019 cm−^3 ;Nv=1× 1019 cm−^3 at 300 K)

EFn = Ec+kBTln

nn 0

Nc

= Ec−(0.026 eV)× 7. 937

= Ec− 0 .206 eV

EFp = Ev−kBTln

pp 0

Nv

= Ev+(0.026 eV)× 2. 3

= Ev+0.06 eV