Conductors in one, two and three dimensions

Charge transfer makes a partly filled band, and that is what conducts

Intuition

A full lift cannot take anyone else. Take a few people out and the lift can move people around again. In solids, an electron-rich or electron-poor partner takes electrons out of (or puts them into) a full band. The band becomes partly filled, so electrons can move. Every example on this page is this trick in a different shape.

Lecture 8 groups conductors by the number of directions in which they conduct. In each case the material is an insulator or a semiconductor until charge transfer gives it a partly filled band. That is the mechanism the online test calls “electron delocalisation and therefore the mechanism for electron conduction (1D/2D/3D)”.

Exam detail

Online Test 1 lists “the nature of electron delocalisation and therefore mechanism for electron conduction in various materials encountered (1D/2D/3D)”. Know each material below, the direction it conducts in, and the source of its partly filled band.

One dimension: Krogmann salts and TTF-TCNQ conduct along chains

In 1842, oxidising K₂[Pt(CN)₄] with chlorine or bromine vapour gave a salt with a metallic sheen. It is called “the first molecular metal”. The 1968 structure shows 1-D chains of face-to-face [Pt(CN)₄]²⁻ units, eclipsed. The oxidised product has chains of face-to-face [Pt(CN)₄]¹·⁷⁻ units, staggered. The conductivity is about 300 S cm⁻¹ along the chains ($\sigma_z$) and about 10⁻³ S cm⁻¹ across them ($\sigma_{x,y}$).

Plain words: the halogen steals some electrons from the platinum chain. A completely filled valence band becomes a partly filled band, which is what a metal has. The slide shows Pt as +2 in the parent and +2.3 in the product.

TTF-TCNQ is “the first organic metal” (1973). Two insulating molecules combine to form 1-D chains. TTF (tetrathiafulvalene) is a good electron donor. TCNQ (7,7,8,8-tetracyano-p-quinodimethane) is a good electron acceptor. The product is (TTF)⁺(TCNQ)⁻. Orbital overlap plus charge transfer from donor to acceptor create a partly filled band and a metallic band structure.

Compare

Krogmann salt

Inorganic. Pt chains. Halogen oxidation empties part of a filled band. First molecular metal (1842 discovery).

TTF-TCNQ

Organic. Donor TTF and acceptor TCNQ stacks. Charge transfer fills part of a band. First organic metal (1973).

The lecture points to Q7 in Tutorial 2 (week 4) for Krogmann salts. The structure figures on these slides did not extract.

Two dimensions: graphite conducts in sheets, and potassium makes it a metal

Graphite uses sp² hybrid orbitals for σ bonds in each carbon sheet. The p_z orbitals above and below the plane overlap to give π bonds, a continuous ring of electron density. That overlap makes a valence band and a conduction band, so conduction happens in 2D, within the carbon planes. Graphite is a semi-metal: thermal energy promotes a small number of electrons into the conduction band.

The numbers: graphite has $\sigma_{x,y} \sim 20{,}000\ \text{S cm}^{-1}$ and $\sigma_z \sim 200\ \text{S cm}^{-1}$. Intercalation means sliding atoms between the sheets. Putting K into graphite gives KC₈, a metal with $\sigma_{x,y} \sim 1{,}000{,}000\ \text{S cm}^{-1}$. Charge moves from K to the carbon sheet and creates a partly filled band.

Optional aside

Lecture 8 shows carbon nanotubes “for interest”. An $(a,b)$ nanotube is a metal if $a = b$, a quasi-metal if $(a - b)$ is a multiple of 3 and $n \neq m$, and otherwise a semiconductor. The slide mixes $(a,b)$ with $n$ and $m$, so the notation is unclear. The (13,0) nanotube is the picture example.

Three dimensions: C₆₀ is a semiconductor and A₃C₆₀ is a metal

Buckminsterfullerene C₆₀ is a football-shaped carbon molecule. Packed into a cubic crystal ($a = 14.17$ Å, space group Fm3̄) it is a semiconductor with a band gap of 1.6 eV. Add alkali metals A between the C₆₀ balls and the crystal turns metallic. Rb₂CsC₆₀ has $a = 14.43$ Å, the same space group, a half-filled $t_{1u}$ band and is a superconductor with $T_c = 31$ K. In it, Rb and Cs sit on octahedral sites and Rb on tetrahedral sites. The slide also mentions a temperature of 350 °C beside the intercalation step.

The molecular orbitals explain why. C₆₀ has 4 × 60 = 240 valence electrons and 90 C-C bonds. 180 electrons fill 90 σ-bonding MOs. The remaining 60 electrons fill 30 π-bonding MOs. The HOMO ($h_u$) and LUMO ($t_{1u}$) are π-type MOs that project outwards and inwards. The symbols come from symmetry and are beyond the scope of the lecture.

In the crystal, outward-projecting MOs of neighbouring C₆₀ molecules overlap and form bands. Charge transfer from the host A to C₆₀ fills the $t_{1u}$ band partway, so A₃C₆₀ is metallic.

Try it: worked example

Count the valence electrons of C60 and split them between sigma and pi MOs.

Show answer and steps

Answer240 in total: 180 in 90 sigma-bonding MOs, 60 in 30 pi-bonding MOs.

1. C has 4 valence electrons and there are 60 atoms: $4 \times 60 = 240$.
2. There are 90 C-C bonds, and each sigma-bonding MO holds 2 electrons: $90 \times 2 = 180$ electrons.
3. Remaining electrons: $240 - 180 = 60$.
4. Each pi-bonding MO holds 2 electrons, so $60 / 2 = 30$ MOs.

Pitfall

Do not say graphite is a metal. The slide calls it a semi-metal, and KC₈ is the metal. Do not say C₆₀ is a metal either: it is a semiconductor until A₃C₆₀ is made.

Recall

What do KC8, A3C60 and TTF-TCNQ have in common?

Charge transfer between two parts of the structure leaves a partly filled band, so they conduct.