Localized versus CollectivedElectrons and Néel Temperatures in Perovskite and Perovskite-Related Structures

It is pointed out that a simple, localized-electron antiferromagnet having one electron per localized orbital interacting via 180\ifmmode^\circ\else\textdegree\fi{} superexchange with similar orbitals on $z$ near-neighbor cations via an overlap integral $\ensuremath{\Delta}$ has a N\'eel temperature $k{T}_{n}\ensuremath{\approx}\frac{z{{q}_{t}b_{t}^{}{}_{}{}^{2}+{q}_{e}b_{e}^{}{}_{}{}^{2}}(S+1)}{S},$ where ${q}_{t}$, ${q}_{e}$ are inversely proportional to electrostatic energies associated with electron transfers; ${b}_{t}\ensuremath{\sim}{\ensuremath{\epsilon}}_{0}{\ensuremath{\Delta}}_{t}$ and ${b}_{e}\ensuremath{\sim}\ensuremath{\epsilon}_{0}^{}{}_{}{}^{\ensuremath{'}}{\ensuremath{\Delta}}_{e}$ are the one-electron transfer integrals of states having ${t}_{2g}$ and ${e}_{g}$ symmetry, respectively. On the other hand, a collective-electron antiferromagnet having a half-filled band that is split in two by the magnetic ordering would have a $k{T}_{N}$ that decreased with increasing bandwidth, and hence increasing $\ensuremath{\Delta}$. This provides a criterion for distinguishing the two cases: $\frac{d{T}_{N}}{\mathrm{dp}}>0$ for localized-electron antiferromagnetism and $\frac{d{T}_{N}}{\mathrm{dp}}<0$ for collective-electron antiferromagnetism, where $p$ is the hydrostatic pressure. It is therefore surprising that $\frac{d{T}_{n}}{d{a}_{0}}>0$, where ${a}_{0}$ is the cation-anion-cation separation, in the systems ${\mathrm{Ca}}_{1\ensuremath{-}x}{\mathrm{Sr}}_{x}\mathrm{Mn}{\mathrm{O}}_{3}$, ${A}^{3+}\mathrm{Fe}{\mathrm{O}}_{3}$, and ${A}^{3+}\mathrm{Cr}{\mathrm{O}}_{3}$, since independent data indicate that the $d$ electrons are localized. This fact is attributed to changes in $A\ensuremath{-}\mathrm{O}$ covalent bonding that cause $\ensuremath{\Delta}$ to increase with the more basic $A$ cation for a given lattice parameter. The relatively small changes in ${T}_{N}$ with $z$ in the series CaMn${\mathrm{O}}_{3}$, ${\mathrm{Ca}}_{4}$${\mathrm{Mn}}_{3}$${\mathrm{O}}_{10}$, ${\mathrm{Ca}}_{3}$${\mathrm{Mn}}_{2}$${\mathrm{O}}_{7}$, and ${\mathrm{Ca}}_{2}$Mn${\mathrm{O}}_{4}$, where geometric considerations alone determine the relative magnitudes of $\ensuremath{\Delta}$, are shown to be consistent with $\frac{d{T}_{N}}{d{a}_{0}}<0$. However, it is pointed out that the magnitude of this geometrical contribution, which should approach a maximum at the localized-electron $\ensuremath{\rightleftarrows}$collective-electron transition, suggests there may be a change from localized $d$ electrons in CaMn${\mathrm{O}}_{3}$, to collective $d$ electrons in ${\mathrm{Ca}}_{2}$Mn${\mathrm{O}}_{4}$. The fact that the paramagnetic susceptibility obeys a Curie-Weiss law in CaMn${\mathrm{O}}_{3}$, but appears to be temperature-independent in ${\mathrm{Ca}}_{2}$Mn${\mathrm{O}}_{4}$, supports this view.