This book (Practical Electron Microscopy and Database) is a reference for TEM and SEM students, operators, engineers, technicians, managers, and researchers.
 

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The superlattices of two perovskites SrTiO₃ and LaTiO₃have many important properties of naturally occurring charge-ordered systems, i.e. mixed-valence configurations near half-filling. Their lattice constants (see SrTiO₃ and LaTiO₃) are relatively well matched, and the continuity of the TiO₆ octahedral lattice across the superlattice minimizes the perturbation of the electronic states near the chemical potential [1-2].

For perovskite cell structures, octahedral oxygen (O) bonding around the atom (Mn for LaMnO₃ as shown in Figure 3518a) in B-site generates the crystal field around the B-site atom. The valence states of the atoms (e.g. Mn) are fixed by charge neutrality. In the case of LaMnO₃, if the parent compound is doped with Sr²⁺ for La³⁺, then Mn³⁺ (d³) is replaced by Mn⁴⁺ in the lattice.

Figure 3518b shows the molecular orbital energy structure for an octahedral TMO₆ cluster. The split t_2g and e_g orbitals can only mix with symmetry adapted linear combinations of the 6 oxygen (O) 2p orbitals having the same symmetry. The separation between the t_2g and e_g levels is called the LF (ligand field) splitting parameter ΔO. σ-overlap is larger than π-overlap so that the upward shift of the e_g level is larger than the upward shift of the t_2g level.

As shown in Figure 3518c, in a cubic transition-metal (TM) oxide crystal, the divalent TM ion is in a site that has octahedral O_h symmetry and the d-levels split into threefold degenerate (lower energy) t_2g states and twofold degenerate (higher energy) e_g states that can accommodate six and four electrons, respectively (including spin states). In this case, these metal d-orbitals are split by an octahedral crystal potential. The tetragonal symmetry splits the levels further. The t_2g states split into a singlet, d_xy, and a doublet d_xz and d_yz. The e_g states split into d_3z2_-r2 and d_x2_-y2 levels.

Figure 3518c. Ligand field splitting of d orbitals in an octahedral ligand field.

Figure 3518d shows the ligand field splitting of d orbitals for Fe in an octahedral ligand field, resulting in lower energy t_2g and higher energy e_g orbitals and presenting the spin states of Fe in octahedral field. Two cases are show on the right of the figure. If an exchange energy larger than the t_2g - e_g splitting favors parallel spin alignment, then a high spin ground state is obtained. If the exchange energy is smaller than the t_2g - e_g splitting, then a low spin ground state is obtained. In this case, the fourth and fifth electrons are not in the high-energy e_g state but are located in the lower t_2g state.

In some TM (transition metal) cases, the filling of orbitals with electrons may affect the local structure and thus induce geometrical distortion around the TM ion. The Jahn–Teller effect, also called Jahn–Teller distortion, describes this type of distortions. A typical Jahn-Teller ion is Mn³⁺ as shown in Figure 3518e. The ion in the high-spin configuration contains a single electron in the upper e_g state when it is placed in an octahedral LF (ligand field). A tetragonal distortion can lower the energy of the system. The lowering in total energy is due to the lowering of one of the e_g orbitals by lengthening the bond along the z axis. Note that the overall energy of the system is not further lowered by splitting the t_2g state because the center of gravity is retained.

[1] Sunstrom, J. E. IV, Kauzlarich, S.M. & Klavins, P. Synthesis, structure, and properties of La1-xSrxTiO3 (0 ≤ x ≤ 1). Chem. Mater. 4, 346–353 (1992).
[2] Kahn, A. H. & Leyendecker, A. J. Electronic energy bands in strontium titanate. Phys. Rev. 135, A1321–A1325 (1964).