28
2 Bonds
(a)
Na
Na
Na
Na
Cl
Cl
Cl
Cl
(b)
Na
Na
Na
Na
Cl
Cl
Cl
Cl
Fig. 2.12 a Experimental and b theoretical charge distribution in the (100) plane of NaCl. The lowest contour in the
interstitial region corresponds to a charge density of 7 e/nm 3 and adjacent contours differ by
√
2. Differences are mainly
due to the fact that the X-ray experiments have been made at room temperature. Adapted from [178]
The charge distribution for NaCl is shown in Fig. 2.12. For tetragonal and orthorhombic structures, the
Madelung constant has been calculated in [179].
2.4 Mixed Bonds
The group-IV crystals are of perfectly covalent nature, the I–VII are almost exclusively ionically
bonded. For III–V (e.g. GaAs, InP) and II–VI compounds (e.g. CdS, ZnO) we have a mixed case.
The (screened) Coulomb potentials of the A and B atoms (in the AB compound) shall be denoted
V A and V B . The origin of the coordinate system is in the center of the A and B atom (i.e. for the
zincblende structure (cf. Sect. 3.4.4) at (1/8, 1/8, 1/8)a. The valence electrons then see the potential
V crystal =
α
V A (r − r α ) +
β
V B (r − r β ) ,
(2.7)
where the sum α (β) runs over all A (B) atoms. This potential can be split into a symmetric (V c ,
covalent) and an antisymmetric (V i , ionic) part (2.8b), i.e. V crystal = V c + V i
V c =
1
2
α
V A (r − r α ) +
α
V B (r − r α )
+
β
V B (r − r β ) +
β
V A (r − r β )
⎫
⎬
⎭
(2.8a)
V i =
1
2
α
V A (r − r α ) −
α
V B (r − r α )
+
β
V B (r − r β ) −
β
V A (r − r β )
⎫
⎬
⎭
.
(2.8b)