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

A comparison between charge-transfer and Mott-Hubbard insulator band structures: a cuprate vs a nickelate.

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

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' ${\displaystyle {\ce {O^2-}}}$ to the ionic ${\displaystyle {\ce {O-}}}$ state.^[1] In this case the ligand hole is often denoted as ${\textstyle {\underline {L}}}$.

Distinguishing between Mott-Hubbard and charge-transfer insulators can be done using the Zaanen-Sawatzky-Allen (ZSA) scheme.^[2]

Exchange interaction

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

${\displaystyle d_{i}^{n}p^{6}d_{j}^{n}\rightarrow d_{i}^{n}p^{5}d_{j}^{n+1}\rightarrow d_{i}^{n-1}p^{6}d_{j}^{n+1}\rightarrow d_{i}^{n}p^{5}d_{j}^{n+1}\rightarrow d_{i}^{n}p^{6}d_{j}^{n}}$.

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

${\displaystyle d_{i}^{n}p^{6}d_{j}^{n}\rightarrow d_{i}^{n}p^{5}d_{j}^{n+1}\rightarrow d_{i}^{n+1}p^{4}d_{j}^{n+1}\rightarrow d_{i}^{n+1}p^{5}d_{j}^{n}\rightarrow d_{i}^{n}p^{6}d_{j}^{n}}$.

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 (${\displaystyle p^{6}\rightarrow p^{5}}$) and two for the second (${\displaystyle p^{6}\rightarrow p^{4}}$).

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.^[1]

References

1. Khomskii, Daniel I. (2014). Transition Metal Compounds. Cambridge: Cambridge University Press. doi:10.1017/cbo9781139096782. ISBN 978-1-107-02017-7.
2. Zaanen, J.; Sawatzky, G. A.; Allen, J. W. (1985-07-22). "Band gaps and electronic structure of transition-metal compounds". Physical Review Letters. 55 (4): 418–421. Bibcode:1985PhRvL..55..418Z. doi:10.1103/PhysRevLett.55.418. hdl:1887/5216. PMID 10032345.