Bonding sets the bands: NaCl, SiC, TiS2 and ReO3

An ionic solid has a full valence band, an empty conduction band and a big gap

Intuition

Think of the ions as full or empty rooms. Chloride has all its outer electrons. Sodium has given its outer electron away. To make current flow you would have to move an electron from a full room to an empty one, and that costs a lot of energy.

Lecture 9 starts with a recap. A solid is largely ionic when it has large cations of low valence (groups I and II) and small electronegative anions (O²⁻, F⁻), as in NaCl. It is largely covalent when it has small, highly polarising cations (B³⁺, Si⁴⁺, P⁵⁺, S⁶⁺) and large, polarisable anions (I⁻, S²⁻), as in SiC.

NaCl is largely ionic. The ions have these configurations: Cl⁻ is 1s²2s²2p⁶3s²3p⁶ and Na⁺ is 1s²2s²2p⁶3s⁰3p⁰. The 3s and 3p orbitals on Cl⁻ overlap to give a fully occupied valence band. The 3s and 3p orbitals on Na⁺ overlap to give an empty conduction band.

Promoting an electron from the valence band to the conduction band is the same as moving charge from Cl⁻ to Na⁺. That is unlikely because of the electronegativity of the ions and because the band gap is large, about 7 eV. The size of the gap correlates with the difference in electronegativity of the ions. The slide’s table (from West) lists gaps such as 8.5 eV for NaCl, 11 eV for LiF and 0.2 eV for InSb.

Exam detail

The slide shows computed band structures of NaCl and SiC “beyond the scope of the course”. The gap table is a trend to read, not a list to memorise. The rule to know: a larger electronegativity difference gives a larger gap.

Similar compounds can sit at opposite ends of the conductivity scale

Lecture 9 gives two pairs where a small change in chemistry flips the behaviour.

Titanium oxide versus sulfide. TiO₂ (rutile) is an insulator, $\sigma \sim 10^{-10}\ \text{S cm}^{-1}$. TiS₂ is metallic because of band overlap, $\sigma \sim 10^{6}\ \text{S cm}^{-1}$.

Rhenium versus tungsten. ReO₃ and WO₃ have the same crystal structure but different electronic structure. In ReO₃ each Re⁶⁺ has one 5d electron. It can delocalise through the band formed by π-type overlap of the $t_{2g}$ orbitals on the metal cation and the p orbitals on oxygen. So ReO₃ is metallic. W⁶⁺ in WO₃ has no 5d electrons, so that band is empty and WO₃ is an insulator.

Compare

ReO3

Re6+ is 5d1. One electron fills part of the band. Metallic.

WO3

W6+ is 5d0. The band is empty. Insulator.

Doping fills the band, so WO₃ can be made metallic

Modifying WO₃ can make it metallic. Doping with Na⁺ gives a tungsten bronze, Na⁺ₓ[W^((6−x)+)O²⁻₃], with $x$ 5d electrons per W cation in the conduction band. The lecture’s summary is: ReO₃ is 5d¹ and metallic, WO₃ is 5d⁰ and insulating, NaWO₃ is 5d¹ and metallic.

A slide with the computed ReO₃ and WO₃ band structures (Materials Project) is also marked beyond the scope.

Pitfall

Same crystal structure does not mean same conductivity. The number of electrons available to fill the band is what differs between ReO₃ and WO₃.

Recall

Plain words: what makes ReO3 conduct and WO3 not?

ReO3 has one electron per metal to put in the band, so the band is partly filled. WO3 has none, so the band is empty.