The early theoretical models

As we have known that, in the 1950s, the theoretical studies of manganites had widely studied on the origin of the ferromagnetic phase. The starting point was that when the colossal magnetoresistance effects were found that promised the potential applications for next generations of electronic devices. In the following, the basic theoretical description for the manganites namely double–exchange and Jahn–Teller effect will be presented.

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2.5.3.1 Double–exchange (DE)

In 1951, Zener presented the basis for understanding the origin of ferromagnetic phase in CMR type materials. After that in a couple of papers [83–84], where mainly qualitative statements and analyses of experiments were also presented and Zener’s work has been widely regarded as providing a proper explanation for ferromagnetism in manganites. Zener predicted ferromagnetism as arising from an indirect coupling between incomplete d-shells, via conducting electrons. It is sufficient to consider a qualitative description of the splitting of the five d-levels in the presence of a 3d orbital crystal environment, as shown in Figure 2.15a). The Hund’s rule for each individual ion or atom enforces a ferromagnetic Hund’s couplingJ_H. This effect was argued by Zener to play an important role in this mechanism, enforcing the configuration, where the unpaired spins are aligned to the lowest energy.

Because the conduction electrons do not change their spin when they move from ion to other ion, the electron interaction or coupling will maintain the z-projection of the spin, Zener gave a reasonable explaination that those electrons are able to move in the crystal in the optimal manner when the net spin of the incomplete d-shells are all parallel. Otherwise, an up electron can land on a down spin ion, and spend an energy proportional to the Hund’s coupling. In the other hand, the conduction electrons have lower their kinetic energy if the background of d-shell spins, or the t_2g spins of manganite, is fully polarized. The kinetic energy is regulated by a hopping amplitude noted as t factor. The d-shell spins are indirectly coupled via an interaction activated by the conduction electrons. Zener clearly remarked in his papers that a direct coupling between d-shells (not mediated by conduction electrons, but by the direct virtual hopping of d-electrons) is of opposite sign leading to antiferromagnetism, rather than to ferromagnetism. The coupling involved in this direct exchange process is calledJ_AF , and it will be shown to be important in the physics of manganites. So, Zener proposed as “double exchange”. This mechanism is sketched in Figure 2.15b). It can be explained as a simultaneous transfer of an electron from the oxygen to the Mn on right side, and from the Mn on left to the oxygen, such that the net transfer is of an electron from Mn on left to Mn on right. This regime leading to ferromagnetism that Zener found should not be confused with that of the super exchange model, which also uses an oxygen as a bridge between ions. He presented that in the super exchange, the interaction leads to an anti-ferromagnetic alignment of spins. Using the double-exchange idea not to explain ferromagnetism but as a tool to

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explain the transfer of electrons that was built in the context of the original idea of ferromagnetism mediated by conduction electrons [83]. Zener’s work was continued by Anderson (Nobel prize 1977) and Hasegawa [85] and De Gennes (Nobel prize 1991) [86], who proposed mechanism in detail. They found out that there is a better way to describe the motion of electrons from Mn-to-Mn and electrons transfer is only one-by-one, still using the oxygen as a bridge between ions, rather than simultaneously as believed by Zener. Any perturbation approach for the hopping amplitude t of electrons will naturally lead to a one-by-one transfer. Perhaps the most often-quoted portion of the work of Anderson and Hasegawa [85] is the effective hopping term t_eff of an electron jumping between two nearest-neighbor Mn ions. In fact, the calculation shows that cos

eff 2

t t  , where  is the angle between t2g spins located at the two sites involved in the electron transfer, as shown in Figure 2.15c).

Figure 2.15: a) Schematic representation of the ideas of Zener to explain ferromagnetism.

Zener envisioned a system with both localized and mobile electrons, which in the manganite language are the t2g and eg electrons, as indicated. b) Schematic view of the DE mechanism.

c) The effective hopping teff mechanism is drawn schematically [82].

2.5.3.2 Jahn–Teller effect

For an isolated 3d ion, five degenerated orbital states are available corresponding to the 3d electrons with l = 2. In a crystal, the degeneracy is partially lifted by the crystal field. The five d- orbitals are split by a cubic crystal field into three t2g orbitals and two eg orbitals. For the MnO₆ octahedron, the splitting between the lowest t_2g level and the highest e_g level is

 1.5eV (Figure 2.16). For the ratio of Mn³⁺ and Mn⁴⁺ ions, the inter atomic correlations ensure parallel alignment of the electron spins (first Hund’s rule); the corresponding exchange energy (This energy will be called later as J_spin ) is estimated about 3 eV [87]. Although the energy of Mn⁴⁺ remains unchanged by such a distortion, the Mn³⁺ has lowered energy. Thus, Mn³⁺ has a marked tendency to distort its octahedral environment in contrast to Mn⁴⁺. This effect is known as Jahn-Teller distortion and it is rather effective in the lightly doped manganites, i.e. with a large concentration, (1 - x), of Mn³⁺ ions. This is illustrated by the structure of LaMnO3 (Figure 2.16) in which the MnO6 octahedra are strongly elongated within the ab-plane in a regular way leading to a doubling of the unit cell. With increasing the Mn³⁺ content, the Jahn-Teller distortions are reduced and the stabilization of the (3z² - r²) eg–orbital becomes less effective. Nevertheless, in a large number of manganites, the e_g–orbitals of two types,









Figure 2.16 Energy level diagram and 3d orbital eigenstates of Mn³⁺ in a crystal field of cubic and tetragonal symmetry [82].

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