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Charge-transfer insulators
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<h2 id="exchange-interaction">Exchange interaction</h2> <p>Analogous to Mott insulators we also have to consider <a href="/facts/Superexchange/8j0aHPEB">superexchange</a> in charge-transfer insulators. One contribution is similar to the Mott case: the hopping of a <i>d</i> electron from one <a href="/facts/Transition_metal/0slGWoLQ">transition metal</a> site to another and then back the same way. This process can be written as </p><p> d i n p 6 d j n → d i n p 5 d j n + 1 → d i n − 1 p 6 d j n + 1 → d i n p 5 d j n + 1 → d i n p 6 d j n {\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}} . </p><p>This will result in an <a href="/facts/Antiferromagnetism/aQ124Kh7">antiferromagnetic</a> exchange (for nondegenerate <i>d</i> levels) with an exchange constant J = J d d {\displaystyle J=J_{dd}} . </p><p> J d d = 2 t d d 2 U d d = 2 t p d 4 Δ C T 2 U d d {\displaystyle J_{dd}={\frac {2t_{dd}^{2}}{U_{dd}}}={\cfrac {2t_{pd}^{4}}{\Delta _{CT}^{2}U_{dd}}}} </p><p>In the charge-transfer insulator case </p><p> d i n p 6 d j n → d i n p 5 d j n + 1 → d i n + 1 p 4 d j n + 1 → d i n + 1 p 5 d j n → d i n p 6 d j n {\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}} . </p><p>This process also yields an antiferromagnetic exchange J p d {\displaystyle J_{pd}} : </p><p> J p d = 4 t p d 4 Δ C T 2 ⋅ ( 2 Δ C T + U p p ) {\displaystyle J_{pd}={\cfrac {4t_{pd}^{4}}{\Delta _{CT}^{2}\cdot \left(2\Delta _{CT}+U_{pp}\right)}}} </p><p>The difference between these two possibilities is the intermediate state, which has one ligand hole for the first exchange ( p 6 → p 5 {\displaystyle p^{6}\rightarrow p^{5}} ) and two for the second ( p 6 → p 4 {\displaystyle p^{6}\rightarrow p^{4}} ). </p><p>The total exchange energy is the sum of both contributions: </p><p> J t o t a l = 2 t p d 4 Δ C T 2 ⋅ ( 1 U d d + 1 Δ C T + 1 2 U p p ) {\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)} . </p><p>Depending on the ratio of U d d and ( Δ C T + 1 2 U p p ) {\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.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> </p> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Khomskii, Daniel I. (2014). Transition Metal Compounds. Cambridge: Cambridge University Press. doi:10.1017/cbo9781139096782. ISBN 978-1-107-02017-7. <a href="978-1-107-02017-7" target="_blank">978-1-107-02017-7</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>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. <a href="https://link.aps.org/doi/10.1103/PhysRevLett.55.418" target="_blank">https://link.aps.org/doi/10.1103/PhysRevLett.55.418</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Khomskii, Daniel I. (2014). Transition Metal Compounds. Cambridge: Cambridge University Press. doi:10.1017/cbo9781139096782. ISBN 978-1-107-02017-7. <a href="978-1-107-02017-7" target="_blank">978-1-107-02017-7</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> </ol>