Charge-transfer insulators

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

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

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

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:

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

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' ${\ce {O^2-}}$ to the ionic ${\ce {O-}}$ state. 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.

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

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

This will result in an antiferromagnetic exchange (for nondegenerate d levels) with an exchange constant $J=J_{dd}$.

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

In the charge-transfer insulator case

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

This process also yields an antiferromagnetic exchange $J_{pd}$:

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

The difference between these two possibilities is the intermediate state, which has one ligand hole for the first exchange ($p^{6}\rightarrow p^{5}$) and two for the second ($p^{6}\rightarrow p^{4}$).

The total exchange energy is the sum of both contributions:

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

Depending on the ratio of $U_{dd}{\text{ and }}\left(\Delta _{CT}+{\tfrac {1}{2}}U_{pp}\right)$, the process is dominated by one of the terms and thus the resulting state is either Mott-Hubbard or charge-transfer insulating.