Metals have different physical properties from either ionic compounds or covalent compounds: high melting and boiling points, hard and crystalline but malleable and ductile, conducts electricity in solid or melt but not soluble nor an electrolyte.
Need a new bonding model in order to account for these properties: use the crystalline structure as a starting point but since we do not have ions we cannot use simple electrostatics; this means that we must use orbitals and overlap as the cohesive force. Consider building a covalent lattice one particle at a time (like we built the ionic lattice):
The orbital diagram is called a quasicontinuum of states or a band. The energy between orbitals is ~ bandwidth/N0 ~ 0 hence called continuous.
A typical metal bandwidth is ~10 eV so the energy between orbitals is
~10 eV/6×1023 ~ 1.7×1023 eV/state or,
in terms of temperature, ~ 1.9×1019 K/state; this means
that electrons can move from orbital to orbital very readily at any temperature.
The bandwidth is related to the amount of overlap: greater overlap leads to a wider band.
What do the orbitals look like?
The lowest energy orbitals are all bonding; the highest are all antibonding. If the band is only partially filled, there is a large bonding energy explaining the high mp and bp.
Metals always have partially filled bands.
Degeneracy of orbitals in the band can exist which leads to the idea of density of states: the number of states found in a given energy interval (mathematically, dN/dE)
Density of States is usually represented graphically as follows:
These represent the solid state equivalent of molecular orbitals
Metallic properties arise from partially filled bands called conduction bands.
Energetically, there are a huge number of bonding orbitals filled and fewer antibonding orbitals so the "bond order" is large which means that metals are hard and have high mp and bp. Mechanical properties: deforming the sample is not traumatic because the electrons can easily move into orbitals that accommodate the changes. Finally, charge transport occurs because an electric field can excite electrons into orbitals delocalized over the whole sample (or at least long distances) which allows motion across the whole sample. All of this depends on the availablity of many (i.e., a high density of states) orbitals at low energy above the highest filled orbital.
The energy of the highest filled orbital (at 0 K) is called the Fermi level; this is equivalent of the chemical potential.
Why is there any electrical resistance?
Defects in the lattice disrupt the delocalization of wavefunctions. If a charge is being moved from one end of a wire to the other end but hits a defect site, the direction of charge motion is changed (i.e., the charge carrier is scattered) and the charge either takes longer to get to the other end or never does - this is resistance. "Defects" can be created by vibrations. If a vibration occurs at the same time that a charge arrives at a particular site, there will be a scattering event. Since the number of vibrations that are active depends on temperature, thus the resistivity must also depend on temperature.
In general, conductivity = = ne, where n is the number of charge carriers, e is the electronic charge, and is the mobility (which is dependent on the number of scattering events)
Heat is carried by both vibrations and electrons. When heat is carried
by electrons, scattering plays a similar role as with conductivity (fewer
scattering events means more thermal conductivity). However, the vibrational
portion requires more vibrations populated to give a higher thermal conductivity.
Thus, the two mechanisms are competing. The net is a product of the two.
Properties: not as hard as metals, lower mp and bp but still solids, not ductile or malleable, poor electrical conductors - in many ways like a bad metal!
How do we explain this in terms of bonding?
Use band theory but with more basis orbitals : require two bands, one
filled (valence band) and one empty (conduction band).
If the band gap is large enough, the material is an insulator because no metal-like properties arise from a filled band.
If the band gap is on the order of thermal energies (small), then excitations of electrons from the valence band into the conduction band occurs.
where (Boltzmann population of the conduction band)
has the same properties as in a metal but the Boltzmann term dominates so the conductivity of a semiconductor increases with increasing temperature
Case 1: the impurity has a filled orbital in the semiconductor band gap
n-type carries are created thermally by excitation from the filled impurity orbital into the conduction band. The population of the valence band is undisturbed; since it is filled no charge can be transported through the valence band. The dopant is a species with more valence electrons than the semiconductor.
This is an n-type semiconductor.
Case 2: the impurity has an empty orbital in the band gap
At T > 0, the valence band is depopulated thermally creating p-type charge carriers (the electrons in the impurity orbital are not moblile, they are too far apart). The dopant must have fewer valence electrons than the semiconductor.
This is a p-type semiconductor
Valence Bond Theory description of semiconductors
n-type: extra electron on the dopant moves onto the host lattice and becomes mobile as a negative charge.
p-type: electron deficient dopant forms a new bond, requiring the use of an electron form the host, leaving a mobile positive charge.
This is a diode. Transistors are similar, but with three junctions (n-p-n or p-n-p)