SEMICONDUCTOR DEVICE PHYSICS AND DESIGN

64 CHAPTER 2. ELECTRONIC LEVELS IN SEMICONDUCTORS

Si

B Negative B ion

Hole

O ne el ectron tak en f rom the

v al ence band to compl ete the

bondi ng of the boron atom hol e

Si

Si

Si

Si

Si

Si

+

A cceptor level

Ea

Eg

E

0

: V al ence band

C onducti on

band

Figure 2.21: Boron has only three valence electrons. It can complete its four fold tetrahedral
bonds only by taking an electron from an Si–Si bond, leaving behind a hole in the silicon valence
band. The positive hole is then available for conduction.

approximation, which gives the relation between the Fermi level and the free carrier concentra-
tion. This approximation is more accurate than the Boltzmann approximation. According to the
Joyce–Dixon approximation , we have

EF=Ec+kBT

[

ln

n

Nc

+

1

√

8

n

Nc

]

=Ev−kBT

[

ln

p

Nv

+

1

√

8

p

Nv

]

(2.8.12)

This relation can be used to obtain the Fermi level ifnis specified. Or else it can be used to
obtainnifEFis known by solving forniteratively.Iftheterm(n/

√

8 Nc)isignored,theresult
correspondstotheBoltzmannapproximation.
If we examine the mobile carrier density dependence upon temperature, there are three regimes,
as shown in figure 2.22 for ann−type material. At low temperatures, the electrons coming from
the donors are attached to the donors and occupy the impurity levelsEd. Thus there is no con-
tribution to the mobile carrier density from the dopants. This regime is called thecarrierfreeze
out regime. At higher temperatures, the dopants ionize until most of them are ionized out over a
temperature regime, the mobile carrier is essentially equal to the dopant density and independent
of temperature. This is the saturation regime and semiconductor devices are operated in this
regime. At very high temperatures, the intrinsic carrier density overwhelms the dopant density
and the material acts as an intrinsic material.
In figure 2.23 we show experimentally measured properties of Mg in GaN (Mg acts as a
deep acceptor in GaN). When the temperature is not extremely high, the hole concentration is
much less than the effective acceptor concentrationNA−ND, since deep acceptors are not fully
ionized at lower temperatures.