SEMICONDUCTOR DEVICE PHYSICS AND DESIGN

150 CHAPTER 4. JUNCTIONS IN SEMICONDUCTORS:P-NDIODES

where the subscriptsnandprefer to then-side andp-side of the device. Using the Boltzmann
approximation for the Fermi level (see equation 2.4.4)

(Ec−EF)n=−kBTln(

nno

Nc

)

wherennois the electron density on then-side of the device. Assuming that all of the donors
are ionized,
nno=Nd

Similarly,
(EF−Ev)p=−kBTln(

pp 0

Nv

)

whereppois the hole density on thep-side and is given by

pp=Na

This gives the built-in voltage

eVbi=Eg+kBTln(

nn 0 pp 0

NcNv

)

Using the relation for intrinsic carrier density

n^2 i=NcNvexp

(

−

Eg

kBT

)

we get

Vbi=

kBT

e

ln(

nn 0 pp 0

n^2 i

) (4.2.3)

Ifnn 0 andnp 0 are the electron densities in then-type andp-type regions, the law of mass action
(i.e., the productnpis constant) tells us that

nn 0 pn 0 =np 0 pp 0 =n^2 i (4.2.4)

This gives for the built-in potential,Vbi=Vn−Vp(figure 4.3)

Vbi=

kBT

e

ln

pp 0

pn 0

(4.2.5)

or

Vbi=

kBT

e

ln

nn 0

np 0

(4.2.6)

We have the following equivalent expressions:

pp 0

pn 0

= exp (eVbi/kBT)=

nn 0

np 0

(4.2.7)