Orbital order

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Figure 1: Schematic representation of orbital ordered states with uniform (left) and alternating (right) occupation of orbitals.

As orbital ordering is called a spatially periodic modulation of the occupation of atomic orbitals by valence electrons in solids (see Figure 1). Such an order pattern can be caused either by interactions between the valence electrons and the crystal lattice or by magnetic correlations and is often associated with direction-dependent ( anisotropic ) magnetic, optical and transport properties.

Cooperative Jahn-Teller effect

In molecules with degenerate valence orbitals, a structural distortion can lower the energy of the ground state. This mechanism, known as the “ Jahn-Teller effect ”, is also effective in solids with degenerate valence orbitals, which can reduce their ground state energy by distorting the unit cell . However, since the distortions of neighboring unit cells are coupled, this “cooperative” Jahn-Teller effect leads to a lowering of the crystal symmetry and to a regularly alternating orbital occupation.

Figure 2: Structure of the MnO 6 octahedra and orbital occupation of lanthanum manganate at temperatures above (above) and below (below) 780 K.

A known example is lanthanum - manganate (LaMnO 3 ) with Mn 3+ ions in the valence electron configuration 3d 4 . The crystal structure of LaMnO 3 is the so-called perovskite structure, in which these ions are surrounded by a cubic octahedron made up of six negatively charged oxygen ions (O 2- ) (see Figure 2 above). By the electrical repulsion between the valence electrons and the O 2 - ligands , the energy levels of the d orbitals split into two groups: The x 2 -y 2 - and 3z 2 -r 2 orbitals in e g group are directly aligned with the neighboring O 2- ions and therefore have a higher energy. The xy, xz and yz orbitals in the t 2g group, on the other hand, are aligned with the gaps so that the distance between the electrons and the O 2 ions is greater and the energy is correspondingly lower. The occupation of these orbitals results from Hund's rules : In the t 2g group, all three orbitals are each occupied by an electron, so that there is no degeneracy. The fourth electron then occupies one of the two degenerate e g orbitals.

Upon cooling, the MnO 6 octahedra distort so that four O 2- ions move closer to the Mn 3+ ion, while the remaining two move further away from it (see Figure 2 below). This tetragonal distortion of the octahedron cancels the degeneracy of the e g orbitals. The energy of the x 2 -y 2 orbital, which is aligned with the closer neighboring O 2- ions, increases. Correspondingly, the energy of the orbital with 3z 2 -r 2 symmetry, which is aligned with the more distant oxygen ions, decreases . The latter orbital is occupied by the electron, while the former remains unoccupied. Since the energy of the ground state is lowered in this way, the octahedron is distorted spontaneously at sufficiently low temperatures. The distortions of neighboring octahedra in the crystal lattice are coupled by the common O 2- ions, so that the elementary cell of the lattice doubles as a result of the distortion and a regular orbital order is created (Figure 3). A structural phase transition into the orbitally ordered phase takes place in LaMnO 3 at 780 K. Orbital order is also observed in doped manganates and other metal oxides such as cobaltates.

Orbital order and magnetism

Figure 3: Orbital order and magnetic order in lanthanum manganate. The green arrows indicate the spins of the manganese ions.

The relative alignment of the valence orbitals on neighboring lattice sites determines the sign and size of the super-exchange interactions , which are responsible for magnetism in magnetic insulators (such as metal oxides). The interplay between orbital order and magnetism was systematically investigated in the 1960s by, among others, John Goodenough , Junjiro Kanamori and Philip W. Anderson and summarized in the so-called "Goodenough-Kanamori-Anderson" rules. The most important rules state that the superexchange interaction in a MOM bond (M = metal ion) is strong and antiferromagnetic if the orbital occupation in the two metal ions is identical. If, on the other hand, orthogonal orbitals are occupied in neighboring metal ions, the interaction is weak and ferromagnetic . In LaMnO 3 , for example, a magnetic phase transition takes place at 140 K. Below this critical temperature, crystal planes with alternating orbital occupation are ferromagnetically ordered (Figure 3). In the direction perpendicular to these planes, on the other hand, the orbital population is uniform and the direction of the magnetic moments alternates.

The Goodenough-Kanamori-Anderson rules apply to compounds in which the electron-lattice interaction is significantly stronger than the superexchange interaction, so that the orbital order takes place at much higher temperatures than the magnetic order. This is especially the case for metal oxides with degenerate e g orbitals, such as LaMnO 3 . In other compounds (such as metal oxides with degenerate t 2g orbitals) the energy scale of both interactions is comparable, so that magnetic correlations can strongly influence the orbital order. A model designed by the physicists Daniel I. Khomskii and Kliment I. Kugel completely neglects the electron-lattice interaction, so that the orbital order comes about exclusively through magnetic correlations.

Individual evidence

  1. ↑ Y. Murakami, JP Hill, D. Gibbs, M. Blume, I. Koyama, M. Tanaka, H. Kawata, T. Arima, Y. Tokura, K. Hirota, Y. Endoh: Resonant X-Ray Scattering from Orbital Ordering in LaMnO 3 . In: Physical Review Letters . tape 81 , no. 3 , July 20, 1998, pp. 582-585 , doi : 10.1103 / PhysRevLett.81.582 ( aps.org [accessed December 29, 2019]).
  2. ↑ Kazuma Hirota, Nobuhisa Kaneko, Akinori Nishizawa, Yasuo Endoh: Two-Dimensional Planar Ferromagnetic Coupling in LaMnO 3 . In: Journal of the Physical Society of Japan . tape 65 , no. 12 , December 15, 1996, ISSN  0031-9015 , p. 3736–3739 , doi : 10.1143 / JPSJ.65.3736 ( jps.jp [accessed December 29, 2019]).
  3. ↑ C. Ulrich, G. Khaliullin, M. Guennou, H. Roth, T. Lorenz, B. Keimer: Spin-Orbital Excitation Continuum and Anomalous Electron-Phonon Interaction in the Mott Insulator LaTiO 3 . In: Physical Review Letters . tape 115 , no. 15 , October 9, 2015, p. 156403 , doi : 10.1103 / PhysRevLett.115.156403 ( aps.org [accessed December 29, 2019]).
  4. ↑ AI ball, DI Khomskii: Crystal structure and magnetic properties of substances with orbital degeneracy . April 1973 (English, jetp.ac.ru [PDF; 309 kB ; accessed on December 29, 2019]).