SEMICONDUCTOR DEVICE PHYSICS AND DESIGN

3.8. CARRIER GENERATION AND RECOMBINATION 125

different hole Fermi level). We then have

n=

∫∞

Ec

Ne(E)fe(E)dE (3.7.1)

p=

∫Ev

−∞

Nh(E)fh(E)dE (3.7.2)

where

fe(E)=

1

exp (E−kBETFn)+1

(3.7.3)

and

fh(E)=1−fv(E)=1−

1

exp (Ek−BETFp)+1

=

1

exp (EFpkB−TE)+1

(3.7.4)

Each band is described by its own Fermi level,EFnandEFp. At equilibriumEFn=EFp.
If excess electrons and holes are injected into the semiconductor, the electron Fermi levelEFn
moves toward the conduction band, while the hole Fermi levelEFpmoves toward the valence
band. This is shown schematically in figure 3.18. By defining separate Fermi levels for the
electrons and holes, one can study the properties of excess carriers using the same relationship
between Fermi level and carrier density as we developed for the equilibrium problem. Thus, in
the Boltzmann approximation we have

n = Ncexp

[

(EFn−Ec)

kBT

]

p = Nvexp

[

(Ev−EFp)

kBT

]

(3.7.5)

For high carrier densities, we have the more accurate Joyce-Dixon approximation:

EFn−Ec = kBT

[

n

n

Nc

+

n

√

8 Nc

]

Ev−EFp = kBT

[

n

p

Nv

+

p

√

8 Nv

]

(3.7.6)

3.8 CARRIER GENERATION AND RECOMBINATION ............

In this section we will examine how mobile carrier densities change when temperature is
changed or light shines on a semiconductor: The electron may start out in the valence band, then