C.6 Electrons in semiconductors 417
valence electrons. Four of them form covalent bonds with the nearest neighbors and the
fifth electron becomes free. If the impurity atom is not ionized, this fifth electron is bound
to the impurity atom, i.e., it rotates around the impurity atom. Thus, an atomic system
very similar to the hydrogen atom is formed: near a positively charged As
+
ion there
rotates one electron. It is easy to remove this electron from this bond, i.e., ionization of
the As atom may be done easily. Such an impurity is called a donor impurity, since the
impurity atom becomes the source (donor) of a free electron.
The impurity atom (As
+
plus an electron) has two substantial differences from the
hydrogen atom, H. First of all, the electron effective mass in a semiconductor, m
n
,isan
order of magnitude smaller than the free-electron mass, m
e
; second, the dielectric constant
of a semiconductor is equal to ≈ 10, while for vacuum this parameter is equal to unity.
Taking into account these peculiarities of an impurity atom and using Eq. (6.7), written
for quantum states of the hydrogen atom, we can find for the energy states of an electron
in an impurity atom
E
(d)
n
=−
k
2
e
m
n
e
4
2
2
h
-
2
1
n
2
. (C.144)
The ionization energy, E
(d)
i
, of an impurity atom is equal to the modulus of the energy of
its ground state (n = 1):
E
(d)
i
=|E
(d)
1
|=−
k
2
e
m
n
e
4
2
2
h
-
2
=
|E
1
|
2
m
n
m
e
, (C.145)
where |E
1
|=13.6 eV is the ionization energy of the hydrogen atom.
Similarly, taking into account Eq. (6.6) for the radius, r
1
, of the hydrogen atom for
the first Bohr orbit, we can write an expression for the size of an impurity atom, i.e., the
average radius of its first Bohr orbit in a crystalline lattice:
r
(d)
i
=
h
-
2
k
e
m
n
e
2
= r
1
m
e
m
n
. (C.146)
Silicon, Si, has a dielectric constant equal to = 12andanelectroneffectivemass
m
n
≈ 0.1m
e
. Thus, the ionization energy of the impurity is less by a factor of
2
m
e
/m
n
≈
1.5 ×10
3
than the ionization energy of the hydrogen atom, i.e., E
(d)
i
≈ 10
−2
eV. Since the
width of the bandgap is about 1 eV, the impurity levels in the bandgap are fairly close to
the bottom of the conduction band. Therefore, these levels are considered “shallow.” The
size of an impurity atom is, according to Eq. (C.146), m
e
/m
n
times larger than the size
of the hydrogen atom, i.e., r
(d)
i
= 100r
1
≈ 5 nm, which is of the order of 10 interatomic
distances in a crystalline lattice.
Let us consider an impurity atom from the third group of the Periodic Table of the
elements (for example, an indium atom, In) in the crystalline lattice of silicon, Si. The
In atom has only three valence electrons, which are not sufficient to form four covalent
bonds with the nearest four Si atoms. This unoccupied bond, i.e., positively charged hole,
may be filled as a result of transition of an electron from a neighboring filled bond. At the
place from which this electron has come, a new hole may be formed, which in its turn can
attract another electron, and so on. Such an impurity is called an acceptor impurity.Asa