SEMICONDUCTOR DEVICE PHYSICS AND DESIGN

60 CHAPTER 2. ELECTRONIC LEVELS IN SEMICONDUCTORS

The effective density of states becomes

Nc =2

(

m∗doskBT

2 π^2

) 3 / 2

=2

(

1. 06 × 0. 91 × 10 −^30 (kg)× 4. 16 × 10 −^21 (J)

2 × 3. 1416 ×(1. 05 × 10 −^34 (Js))^2

) 3 / 2

m−^3

=2. 78 × 1025 m−^3 =2. 78 × 1019 cm−^3

We can see the large difference in the effective density between Si and GaAs.
In the case of the valence band, we have the heavy hole and light hole bands, both of which contribute
to the effective density. The effective density is

Nv=2

(

m^3 hh/^2 +m^3 h/^2

)(kBT

2 π^2

) 3 / 2

For GaAs we usemhh=0. 45 m 0 ,mh=0. 08 m 0 and for Si we usemhh=0. 5 m 0 ,mh=0. 15 m 0 ,to
get

Nv(GaAs) = 7. 72 × 1018 cm−^3

Nv(Si) = 9. 84 × 1018 cm−^3

2.8 DOPING OF SEMICONDUCTORS

To avoid leakage current in the ‘OFF’ state, semiconductor devices operate at temperatures
where the intrinsic carrier density is small(∼< 1015 cm−^3 ). To introduce electrons and holes in a
semiconductor the material is doped with dopants. The electrons (holes) created by the dopants
areusedindevicedesign.
Donors are dopants which can donate an electron to the conduction band and acceptors are
dopants which can accept an electron from the valence band and thus create a hole. The donor
atom replaces a host atom in the crystal and contains one (or more) extra electrons in its outer
shell. The donor atom could be a pentavalent atom in Si or a Si atom on a Ga site in GaAs.
Focusing on the pentavalent atom in Si, four of the valence electrons of the donor atom behave
as they would in a Si atom; the remaining fifth electron now sees a positively charged ion to
which it is attracted, as shown in figure 2.19. The ion has a charge of unity and the attraction is
simply Coulombic suppressed by the dielectric constant of the material. The problem is now that
of the hydrogen atom case, except that the electron mass is the effective mass at the bandedge.
The attractive potential is

U(r)=

−e^2

4 πr

(2.8.1)

whereis the dielectric constant of the semiconductor; i.e., the product of 0 and the relative
dielectric constant. In this simplification the properties of the dopant atom can be described
by a simple hydrogen-like model, where theelectronmassissimplytheeffectivemassatthe
bandedge.