SEMICONDUCTOR DEVICE PHYSICS AND DESIGN

62 CHAPTER 2. ELECTRONIC LEVELS IN SEMICONDUCTORS

Si

As

Positive

ion

E xcess

el ectron

+ Ec Ed: Donor level

E

Ev Valence

band

C onducti on

band

A rseni c ( A s) atom donates one

el ectron to the conducti on band to

pr oduc e an n-type silicon

Si

Si

Si

Si

Si

Si

Figure 2.20: A schematic of doping of Si with arsenic (or other group V dopant). A donor level
is produced below the conduction bandedge.

Note that in the hydrogen atom problem the electron levels are measured from the vacuum energy
level which is taken asE=0.Inthedonorproblem,theenergylevelismeasuredfromthe
bandedge. Figure 2.20 shows the energy level associated with a donor impurity.
The wavefunction of the ground state is as in the hydrogen atom problem

Fc(r)=

1

√

πa^3

e−r/a (2.8.5)

whereais the donor Bohr radius and is given by

a=

(4π)^2

m∗ee^2

=0. 53

(

/ 0

m∗e/m 0

)

A ̊ (2.8.6)

For most semiconductors the donor energies are a few meVs below the conduction bandedge
and the Bohr radius is∼ 100 A. ̊
Note that donors are defect levels, which are neutral when an electron occupies the defect
level and positively charged when unoccupied. Acceptors are neutral when empty and negatively
charged when occupied by an electron. The acceptor levels are produced when impurities, which
have a similar core potential as the atoms in the host lattice, but have one less electron in the
outermost shell, are introduced into the crystal.
As shown in figure 2.21 the acceptor impurity potential could now be considered to be equiv-
alent to a host atom potential, together with the Coulombic potential of a negatively charged
particle. The “hole” (i.e., the absence of an electron in the valence band) can then bind to the ac-
ceptor potential. The effective mass equation can again be used, since only the top of the valence
band contributes to the acceptor level. The valence band problem is considerably more complex