The Mott-Hubbard Insulating State

The Hubbard picture is the most celebrated and simplest model of the Mott insulator. It is comprised of a tight-binding Hamiltonian, written in the second quantization formalism. Second quantization is the name given to the quantum field theory procedure by which one moves from dealing with a set of particles to a field. Quantum field theory is the study of the quantum mechanical interaction of elementary particles with fields. Quantum field theory is such a notoriously difficult subject that this textbook will not attempt to go beyond the level of merely quoting equations. The Hubbard Hamiltonian is  

Equation 7.2 contains just two parameters t, the transfer integral and U, which represents the Coulomb repulsion of two antiparallel spin electrons at the same site. It is possible to 

The distance between two electrons at a given site is given as ri2. The electron wave function for one of the electrons is given as (p(ri) and the wave function for the second electron, with antiparallel spin, is p(r2)- Equation 7.3 is called the Hubbard intra-atomic energy and it is not accounted for in conventional band theory, in which the independent electron approximation is invoked. Finally, it should also be noted that the Coulomb repulsion interaction had been introduced earlier in the Anderson model describing a magnetic impurity coupled to a conduction band (Anderson, 1961). In fact, it has been shown that the Hubbard Hamiltonian reduces to the Anderson model in the limit of infinite-dimensional (Hilbert) space (Izyumov, 1995). Hence, Eq. 7.3 is sometimes referred to as the Anderson-Hubbard repulsion term. 

As alluded to in Section 5.5.5, the transition metal monoxides with the rock-salt strucmre are the archtypical examples of correlated systems. Of these oxides, only TiO is metallic. The others, NiO, CoO, MnO, FeO, and VO, are all insulating, despite the fact that the Fermi level falls within a partially filled band (in the independent electron picture). Direct electron transfer between two of the transition metal cations (in the rock-salt strucmre, d d interactions are important owing to the proximity of the cations), say, manganese, is equivalent to the disproportionation reaction  

It costs energy, U, to transfer an electron from one Mn into the aheady occupied d orbital of another Mn ion d d ), thereby, opening a Coulomb gap. The d band 

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It has been seen in the previous section that the ratio of the onsite electron-electron Coulomb repulsion and the one-electron bandwidth is a critical parameter. The Mott-Hubbard insulating state is observed when U > W, that is, with narrow-band systems like transition metal compounds. Disorder is another condition that localizes charge carriers. In crystalline solids, there are several possible types of disorder. One kind arises from the random placement of impurity atoms in lattice sites or interstitial sites. The term Anderson localization is applied to systems in which the charge carriers are localized by this type of disorder. Anderson localization is important in a wide range of materials, from phosphorus-doped silicon to the perovskite oxide strontium-doped lanthanum vanadate, Lai cSr t V03. 

Several physical parameters are summarized in Table 10.2. Since the bandwidths are comparable with or less than the effective on-site Coulombic repulsion, they are in the proximity of the Mott-Hubbard insulating state. The conduction electrons are strongly correlated in the metallic state to give rise to curious phenomena. As a result, there are many physical properties which are not fully understood yet. 

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