Introduction to the Physics of Electrons in Solids by Henri Alloul

Author:Henri Alloul
Language: eng
Format: epub
Publisher: Springer Berlin Heidelberg, Berlin, Heidelberg

8.9.1 Mott–Hubbard Insulator

The very existence of magnetic insulators raises a fundamental problem for the band theory. According to the discussion in Chap. 3, all insulators should be non-magnetic (or more precisely, slightly diamagnetic). In order to explain the magnetic behaviour discussed in this chapter, we must therefore reassess the underlying approximations that led to the band theory, and especially the averaging approach to the Coulomb interactions between electrons.

Let us begin by considering the case of an isolated atom (on the left in Fig. 8.13), which was the starting point for the tight-binding theory in Chap. 1. In this context, it was assumed that the energy brought to the system by an extra electron would be ε 0, and that a second electron on the same atom would also bring ε 0, so that the total energy would be for a doubly negatively charged ion. But this is obviously not very realistic, owing to Coulomb repulsion. Apart from its ‘orbital’ energy ε 0, the second electron will also be subject to the Coulomb repulsion of the first electron, and its energy will thus be higher than ε 0 by an amount usually denoted by U, which represents the Coulomb repulsion between the first and second electrons added to an initially neutral atom. The total energy of the doubly negative ion is thus . Note that U can vary considerably depending on the atom (from about 1 eV to more than 10 eV). This very simple approach goes by the name of the Hubbard model.

Fig. 8.13Level diagram for the Mott–Hubbard model. Left: Isolated atom. Center: Mott–Hubbard insulator obtained for a small hopping integral . Right: Metallic situation corresponding to

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