SEMICONDUCTOR DEVICE PHYSICS AND DESIGN

5.6. SEMICONDUCTOR HETEROJUNCTIONS 233

Ec

eVd1

-Wn 0 Wp

eVd2

Ev

EF

ΔEc

ΔEv

eΧ 1 eΧ 2

Eg1 Eg2

Ec1

EF1

Ev1

Evac

Ec2

EF2

Ev2

(a) (b)

Figure 5.9: (a) Band line-ups of two distinct materials prior to the formation of a junction. (b)
Band diagram of a heterojunction formed between the two materials.

term. Making the same substitutions for(EF−Ev)pand(Ec−EF)nas we made in thep-n
homojunction case gives us the built-in potential as

Vbi=

1

e

(ΔEc+Eg 2 )−

kBT

e

n

[

Nc 1 Nv 2

n 1 p 2

]

(5.6.2)

whereNc 1 is the conduction band density of states in material 1,Nv 2 is the valence band den-
sity of states in material 2,n 1 =Nd 1 is the electron concentration in material 1 (assuming full
ionization), andp 2 =Na 2 is the hole concentration in material 2 (also assuming full ionization).
The depletion region widthW(Vbi)and the electric field profile in the depletion region can
be found in the same way as for ap-nhomojunction, except that 1 = 2 , so the electric field
is not continuous at the material interface. The charge density and electric field profiles in the
structure are shown in figure 5.10. Gauss’ Law states that the displacement fieldD=E(Eis
the electric field) must be continuous at the interface. This gives the relationship

 1 E 1 ,m= 2 E 2 ,m (5.6.3)

whereE 1 ,mis the maximum electric field in material 1 andE 2 ,mis the maximum electric field
in material 2 (see figure 5.10).