Charge-transfer insulators

Charge-transfer insulators are a class of materials predicted to be conductors following conventional band theory, but which are in fact insulators due to a charge-transfer process.

Unlike in Mott insulators, where the insulating properties arise from electrons hopping between unit cells, the electrons in charge-transfer insulators move between atoms within the unit cell.

In the Mott–Hubbard case, it's easier for electrons to transfer between two adjacent metal sites (on-site Coulomb interaction U); here we have an excitation corresponding to the Coulomb energy U with

{\displaystyle d^{n}d^{n}\rightarrow d^{n-1}d^{n+1},\quad \Delta E=U=U_{dd}}

In the charge-transfer case, the excitation happens from the anion (e.g., oxygen) p level to the metal d level with the charge-transfer energy Δ:

{\displaystyle d^{n}p^{6}\rightarrow d^{n+1}p^{5},\quad \Delta E=\Delta _{CT}}

U is determined by repulsive/exchange effects between the cation valence electrons.

Δ is tuned by the chemistry between the cation and anion.

One important difference is the creation of an oxygen p hole, corresponding to the change from a 'normal'

to the ionic

{\displaystyle {\ce {O-}}}

state.

[1] In this case the ligand hole is often denoted as

Distinguishing between Mott-Hubbard and charge-transfer insulators can be done using the Zaanen-Sawatzky-Allen (ZSA) scheme.

[2] Analogous to Mott insulators we also have to consider superexchange in charge-transfer insulators.

One contribution is similar to the Mott case: the hopping of a d electron from one transition metal site to another and then back the same way.

This process can be written as

This will result in an antiferromagnetic exchange (for nondegenerate d levels) with an exchange constant

{\displaystyle J=J_{dd}}

{\displaystyle J_{dd}={\frac {2t_{dd}^{2}}{U_{dd}}}={\cfrac {2t_{pd}^{4}}{\Delta _{CT}^{2}U_{dd}}}}

In the charge-transfer insulator case

This process also yields an antiferromagnetic exchange

{\displaystyle J_{pd}}

{\displaystyle J_{pd}={\cfrac {4t_{pd}^{4}}{\Delta _{CT}^{2}\cdot \left(2\Delta _{CT}+U_{pp}\right)}}}

The difference between these two possibilities is the intermediate state, which has one ligand hole for the first exchange (

The total exchange energy is the sum of both contributions:

{\displaystyle J_{total}={\cfrac {2t_{pd}^{4}}{\Delta _{CT}^{2}}}\cdot \left({\cfrac {1}{U_{dd}}}+{\cfrac {1}{\Delta _{CT}+{\tfrac {1}{2}}U_{pp}}}\right)}

Depending on the ratio of

{\displaystyle U_{dd}{\text{ and }}\left(\Delta _{CT}+{\tfrac {1}{2}}U_{pp}\right)}

, the process is dominated by one of the terms and thus the resulting state is either Mott-Hubbard or charge-transfer insulating.