Page 116 - Materials Chemistry, Second Edition

103
2.3. The Crystalline State

2.3.7. Bonding in Crystalline Solids: Introduction to Band Theory
Building on the concept of ‘reciprocal space’ and appreciation of some physical
properties of crystals, we are now in a position to discuss how the bonding motif of a
metallic or semiconductor crystal affects its electrical conductivity. Chemists are
familiar with traditional molecular orbital diagrams that are comprised of linear
combinations of atomic orbitals (L.C.A.O. – M.O. theory). For diatomic molecules,
the overlap of two energetically-similar s-orbitals results in s (bonding) and s*
(antibonding) molecular orbitals. The overlap of p-orbitals may result in s/s* (via
overlap of p z atomic orbitals), as well as p/p* orbitals via p x and p y interactions.
The overlap for metals containing d-orbitals is more complex, which gives rise to s,
p and d bonding/antibonding molecular orbitals (Figure 2.67).
A key concept in LCAO-MO bonding theory is the formation of the same
number of molecular orbitals as the number of atomic orbitals that are combined
(e.g., there are 12 M.O.s formed when 4s and 3d atomic orbitals combine in the
Ti 2 molecule, see Figure 2.68a). As the number of atoms increases to infinity
within a crystal lattice, the DE 0 between energy levels within bonding
and antibonding regions (Figure 2.68b). This is an application of the Pauli exclu-
sion principle, which states for N electrons, there must be N/2 available states to
house the electron density. [57]
The electron-occupied band is known as the valence band, whereas the unfilled
band is referred to as the conduction band. The energy gap, if present, between these

a σ *
+
d z 2 d z 2 σ
b
π*
+

d yz d yz π
or or
d xz d xz

c δ *
+

δ
2 2
2 2
d - y d - y
x
x
or or
d xy d xy
Figure 2.67. Illustration of d-orbital overlap between adjacent metal atoms in an extended metallic
network.