182
6 Magnetism and Conduction
and t
−
ab can again be calculated from the energy difference of the two doublets, see
Eq. 6.10:
t
−
ab =
1
2
∆E 12 ==abb| ˆ
H|aab
(6.13)
6.2 Double Exchange
The concept of double exchange was introduced by Zener in the 1950s to explain the
sudden drop in the resistance of certain manganites when an external magnetic field
was applied [2]. The manganese ions in these compounds have either three or four
unpaired electrons in the 3d-shell; three electrons in t 2g -like orbitals and the fourth
electron in an e g -like orbital. Electron hopping in such compounds takes place in the
presence of other unpaired electrons and depending on the inter-site spin coupling of
the t 2g electrons, the e g electron has more or less probability to hop to a neighboring
site. Assume that the antiferromagnetic coupling dominates in the absence of an
external magnetic field. As shown in the upper part of Fig. 6.2, the electron hopping
leads to a state that does not have the maximum spin on the center that receives
the electron, whereas maximum spin coupling at one site is preferred as stated by
Hund’s rule. By applying a sufficiently large external magnetic field, the spins on all
centers can be forced in a ferromagnetic alignment. In such a case, see the lower part
of the figure, the electron hopping creates a high-spin state on the receiving center,
which corresponds to the local ground state and this is more favorable for electron
mobility, i.e. an external magnetic field can drastically lower the resistance to electric
conductance.
The effectiveness of the electron hopping between two metal centers separated by
a closed-shell ion, typically O 2− , inspired Zener to introduce the concept of double
exchange illustrated in Fig. 6.3. The simultaneous hopping process of an electron with
α-spin from the first metal center to the O 2− ion and from this O 2− ion to the second
metal center was held responsible for the hopping. Contrary to the superexchange
described in Fig. 5.4, the double exchange only involves electrons of the same spin.
Therefore, the intuitive picture is that since the electron transfers can take place
simultaneously, in contrast to the superexchange, the double exchange hopping is
very efficient.
This simple electron hopping explanation has later been revised to incorporate
the strong electron-phonon coupling caused by the Jahn-Teller splitting of the Mn 3+
ions. The conduction is due to the hopping of a magnetic polaron rather than a bare
electron [3].
6 Magnetism and Conduction
and t
−
ab can again be calculated from the energy difference of the two doublets, see
Eq. 6.10:
t
−
ab =
1
2
∆E 12 ==abb| ˆ
H|aab
(6.13)
6.2 Double Exchange
The concept of double exchange was introduced by Zener in the 1950s to explain the
sudden drop in the resistance of certain manganites when an external magnetic field
was applied [2]. The manganese ions in these compounds have either three or four
unpaired electrons in the 3d-shell; three electrons in t 2g -like orbitals and the fourth
electron in an e g -like orbital. Electron hopping in such compounds takes place in the
presence of other unpaired electrons and depending on the inter-site spin coupling of
the t 2g electrons, the e g electron has more or less probability to hop to a neighboring
site. Assume that the antiferromagnetic coupling dominates in the absence of an
external magnetic field. As shown in the upper part of Fig. 6.2, the electron hopping
leads to a state that does not have the maximum spin on the center that receives
the electron, whereas maximum spin coupling at one site is preferred as stated by
Hund’s rule. By applying a sufficiently large external magnetic field, the spins on all
centers can be forced in a ferromagnetic alignment. In such a case, see the lower part
of the figure, the electron hopping creates a high-spin state on the receiving center,
which corresponds to the local ground state and this is more favorable for electron
mobility, i.e. an external magnetic field can drastically lower the resistance to electric
conductance.
The effectiveness of the electron hopping between two metal centers separated by
a closed-shell ion, typically O 2− , inspired Zener to introduce the concept of double
exchange illustrated in Fig. 6.3. The simultaneous hopping process of an electron with
α-spin from the first metal center to the O 2− ion and from this O 2− ion to the second
metal center was held responsible for the hopping. Contrary to the superexchange
described in Fig. 5.4, the double exchange only involves electrons of the same spin.
Therefore, the intuitive picture is that since the electron transfers can take place
simultaneously, in contrast to the superexchange, the double exchange hopping is
very efficient.
This simple electron hopping explanation has later been revised to incorporate
the strong electron-phonon coupling caused by the Jahn-Teller splitting of the Mn 3+
ions. The conduction is due to the hopping of a magnetic polaron rather than a bare
electron [3].