2.4 Mixed Bonds
29
(a)
(b)
Fig. 2.13 Schematic representation of a bonding and b antibinding sp 3 orbitals. From [169]
For homopolar bonds V i = 0 and the splitting between bonding and antibinding states is E h , which
mainly depends on the bond length l AB (and the related overlap of atomic wavefunctions). In a partially
ionic bond the orbitals are not symmetric along A–B, but the center is shifted towards the more
electronegative material (Figs. 2.7c, d and 2.13).
The band splitting
4 between the (highest) bonding and (lowest) antibinding state E ba is then written as
E ba = E h + iC ,
(2.9)
where C denotes the band splitting due to the ionic part of the potential and depends only on V A − V B .
C is proportional to the difference of the electronegativities X of the A and B atoms, C(A, B) =
5.75(X A − X B ). A material thus takes a point in the (E h ,C) plane (Fig. 2.14). The absolute value for
the band splitting is given as E
2
ba = E
2
h + C
2 .
The ionicity of the bond is described with the ionicity (after Phillips) f i , defined as [181, 182]
f i =
C
2
E
2
h + C 2 .
(2.10)
The covalent part is 1 − f i . In Table 2.1 the ionicity is given for a number of binary compounds. The
ionicity can also be interpreted as the angle tan(φ) = C/E h in the (E h ,C) diagram. The critical value
of f i = 0.785 for the ionicity separates quite exactly (for about 70 compounds) the 4-fold (diamond,
zincblende and wurtzite) from the 6-fold (rocksalt) coordinated substances ( f i = 0.785 is indicated
by a dashed line in Fig. 2.14).
For ionic compounds, an effective ionic charge e
∗ is defined connecting the displacement u of
negative and positive ions and the resulting polarization P = (e
∗
/2a
3
) u [183]. Connected with the
ionicity is the so-called s-parameter, describing the change of the charge upon change of bond length
b from its equilibrium value b 0 [184]
e
∗
(b) = e
∗
(b 0 )
b
b 0
s
≈ e
∗
0 (1 + s ) ,
(2.11)
being the strain of the bond length, b/b 0 = 1 + . It seems justified to assume that e
∗
(b 0 ) is always
positive at the metal atom in III–V and II–VI compounds. The relation of s with the ionicity f i is shown
in Fig. 2.15 for various compound semiconductors.
4 This energy should not be confused with the band gap cv , the energy separation of the highest valence-band state and
the lowest conduction-band state. The energy splitting E ba is the energy separation between the centers of the valence
and conduction bands. Mostly, the term E g is used for cv .
29
(a)
(b)
Fig. 2.13 Schematic representation of a bonding and b antibinding sp 3 orbitals. From [169]
For homopolar bonds V i = 0 and the splitting between bonding and antibinding states is E h , which
mainly depends on the bond length l AB (and the related overlap of atomic wavefunctions). In a partially
ionic bond the orbitals are not symmetric along A–B, but the center is shifted towards the more
electronegative material (Figs. 2.7c, d and 2.13).
The band splitting
4 between the (highest) bonding and (lowest) antibinding state E ba is then written as
E ba = E h + iC ,
(2.9)
where C denotes the band splitting due to the ionic part of the potential and depends only on V A − V B .
C is proportional to the difference of the electronegativities X of the A and B atoms, C(A, B) =
5.75(X A − X B ). A material thus takes a point in the (E h ,C) plane (Fig. 2.14). The absolute value for
the band splitting is given as E
2
ba = E
2
h + C
2 .
The ionicity of the bond is described with the ionicity (after Phillips) f i , defined as [181, 182]
f i =
C
2
E
2
h + C 2 .
(2.10)
The covalent part is 1 − f i . In Table 2.1 the ionicity is given for a number of binary compounds. The
ionicity can also be interpreted as the angle tan(φ) = C/E h in the (E h ,C) diagram. The critical value
of f i = 0.785 for the ionicity separates quite exactly (for about 70 compounds) the 4-fold (diamond,
zincblende and wurtzite) from the 6-fold (rocksalt) coordinated substances ( f i = 0.785 is indicated
by a dashed line in Fig. 2.14).
For ionic compounds, an effective ionic charge e
∗ is defined connecting the displacement u of
negative and positive ions and the resulting polarization P = (e
∗
/2a
3
) u [183]. Connected with the
ionicity is the so-called s-parameter, describing the change of the charge upon change of bond length
b from its equilibrium value b 0 [184]
e
∗
(b) = e
∗
(b 0 )
b
b 0
s
≈ e
∗
0 (1 + s ) ,
(2.11)
being the strain of the bond length, b/b 0 = 1 + . It seems justified to assume that e
∗
(b 0 ) is always
positive at the metal atom in III–V and II–VI compounds. The relation of s with the ionicity f i is shown
in Fig. 2.15 for various compound semiconductors.
4 This energy should not be confused with the band gap cv , the energy separation of the highest valence-band state and
the lowest conduction-band state. The energy splitting E ba is the energy separation between the centers of the valence
and conduction bands. Mostly, the term E g is used for cv .