Page 307 - Instant notes

Strong solid phase interactions 293

atomic orbitals; (b) metallic conductor
with partially filled band resulting
from band overlap; (c) an insulator;
(d) a semiconductor.

band) by a band gap (Fig. 4c). The band gap is large enough to prevent thermal
excitation of electrons into the conduction band. Electrical conduction is prevented
because the electrons do not have vacant energy levels into which they may migrate, and
so cannot carry charge through the material.
A semiconductor has a similar overall band structure to that of an insulator, and there
is no clear distinction between the two. What difference there is lies in the size of the
band gap, which is small enough in a semiconductor to allow thermal excitation of
electrons into the conduction band (Fig. 4d). For materials which are typically regarded
as semiconductors at room temperature (silicon, gallium arsenide, gallium nitride, for
example), the band gap is of the order of 1–3 eV. Raising the temperature of a
semiconductor increases the number of electrons promoted across the band gap.
The conductivity of a semiconductor increases with temperature as more electrons are
promoted into the conduction band; this behavior contrasts with the behavior of a metal,
whose conductivity decreases with temperature. Metallic behavior is explained by the
increased thermal motion of the lattice, which limits the electron mobility by increasing
the electron-lattice collision frequency.

Coulombic effects in the solid phase

In ionic solids, ions are primarily held together by electrostatic forces. The strength of the
binding in these solids is measured in terms of the energy required to fully dissociate the
ions in the lattice into gaseous ions. The energy required for this process to be brought
about is the lattice enthalpy (see Topic B3).
If e is the charge on an electron, and the number of charges on a pair of ions, A and B,
held a distance r AB apart are z A and z B respectively, then the electrostatic bonding energy
between the pair may be calculated:

where ε 0 is the vacuum permittivity. The electrostatic energy of an ion in an infinite one-
dimensional array of ions may be easily calculated by summation of the individual terms,
but calculation of the potential energy in higher-dimensional lattices is mathematically
challenging. In order to calculate the total coulombic potential for an ion, the contribution
from every other ion is required. The importance of the strength of each interaction
decreases with distance, but the summation is made difficult because the number of
interactions increases with distance. It is found that, for regular crystals, the molar
potential energy is given by: