Charge-transfer insulators explained

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

d ^{nd} ^{n} → d ^{n-1} d ^{n+1} , \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:

d ^{np} ^{6} → d ^{n+1} p ^{5} , \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' O^2- to the ionic O- state.^[1] In this case the ligand hole is often denoted as $\backslash underline$.

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

d ^{6 d} _{ip} ^{n} ^{j} → d ^{5 d} _{ip} ^{n +1} ^{j} → d ^{6 d} _{ip} ^{n +1} ^{j} → d ^{5 d} _{ip} ^{n +1} ^{j} → d ^{6 d} _{ip} ^{n} ^{j} .

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

J = J _{dd} .J _{dd} = (2 t ^{2} _{dd})/(U _{dd}) = \cfrac{ 2 t ^{4} _{pd} }

In the charge-transfer insulator case

d ^{n} _{i} p ^{6d} ^{n} ^{j} → d ^{n} _{i} p ^{5d} ^{n +1} ^{j →} d ^{n +1} _{i} p ^{4d} ^{n +1} ^{j →} d ^{n +1} _{i} p ^{5d} ^{n} ^{j →} d ^{n} _{i} p ^{6d} ^{n} ^{j} .

This process also yields an antiferromagnetic exchange

J _{pd} :J _{pd} = \cfrac{ 4 t ^{4} _{pd} }

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

p ^{6 →} p ^{5} ) and two for the second (p ^{6 →} p ^{4} ).

The total exchange energy is the sum of both contributions:

J _{total} = \cfrac{ 2 t ^{4} _{pd} } \cdot \left(\cfrac + \cfrac\right).

Depending on the ratio of

U _{dd} 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.

References

1. Book: Khomskii, Daniel I.. Transition Metal Compounds. 2014. Cambridge University Press. 978-1-107-02017-7. Cambridge. 10.1017/cbo9781139096782.
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. 10.1103/PhysRevLett.55.418. 10032345 . 1985PhRvL..55..418Z . 1887/5216. free.