Mott-Hubbard gap

This signals the presence of a gap Ap-Tp in the charge degrees of freedom, which has the same origin as the Mott-Hubbard gap in the one-dimensional Hubbard model at half-filling [119]. 

Despite the success of this simple picture, not all B-site cations with partially filled d orbitals give rise to metallic perovskites, and many are antiferromagnetic (AFM) insulators. This dilemma was resolved in the middle of the twentieth century with respect to transition metal oxides by assuming that electron correlation, taking the form of Coulomb repulsion between electrons, split the partly filled 3d band into a filled and an empty sub-band, with a band gap between the two. This gap is the Hubbard or Mott-Hubbard gap, denoted by U, and the compounds that display this type of behaviour are called Mott insulators. 

As shown in the previous sections, diagonalization of the VBCI matrices involves the CT energies A, transfer integrals (or and Mott-Hubbard gaps U. While experimental information on... 

This is termed the Mott-Hubbard gap. This energy denoted by is of the order of the Coulomb interaction energy U. (ii) One electron that is removed from a O site is added to the M site, that is, d"p This is termed the charge-... 

The AM4Q8 (A = Ga, Ge M = V, Nb, Ta Q = S, Se) compounds exhibit a lacunar spinel structure [8] with tetrahedral transition metal clusters M4 (see Figure la). These compounds are Mott insulators exhibiting a very small Mott-Hubbard gap (0.2 0.1 eV) due to the presence of the M4 clusters [9]. A direct consequence of this low gap value is a high sensitivity to external perturbations such as doping or external pressure which can induce an insulator to metal transition in these compounds [10,11,12,13]. We have recently shown that these compounds are also very sensitive to electric pulses [3,4,14,15]. The AM4Q8 (A = Ga, Ge M = V, Nb, Ta Q = S, Se) Mott insulator compounds exhibit indeed an unprecedented type of resistive switching (RS) of interest for... 

First we consider the origin of band gaps and characters of the valence and conduction electron states in 3d transition-metal compounds [104]. Band structure calculations using effective one-particle potentials predict often either metallic behavior or gaps which are much too small. This is due to the fact that the electron-electron interactions are underestimated. In the Mott-Hubbard theory excited states which are essentially MMCT states are taken into account dfd -y The subscripts i and] label the transition-metal sites. These... 

When the cores are approached, the sub-bands split, acquiring a bandwidth, and decreasing the gap between them (Fig. 14 a). At a definite inter-core distance, the subbands cross and merge into the non-polarized narrow band. At this critical distance a, the narrow band has a metallic behaviour. At the system transits from insulator to metallic (Mott-Hubbard transition). Since some electrons may acquire the energies of the higher sub-band, in the solid there will be excessively filled cores containing two antiparallel spins and excessively depleted cores without any spins (polar states). 

Hubbard (13) elucidated a mathematical description of the change from one situation to another for the simplest case of a half-filled s band of a solid. His result is shown in Figure 11. For ratios of W/U greater than the critical value of 2/ /3 then a Fermi surface should be found and the system can be a metal. This critical point is associated with the Mott transition from metal to insulator. At smaller values than this parameter, then, a correlation, or Hubbard, gap exists and the system is an antiferromagnetic insulator. Both the undoped 2-1 -4 compound and the nickel analog of the one dimensional platinum chain are systems of this type. At the far left-hand side of Figure 11 we show pictorially the orbital occupancy of the upper and lower Hubbard bands. 

It can be seen that e2 drops linearly at first, but has lower slope near the transition. There is no discontinuity, as would be expected for a Mott transition in a crystal (Chapter 4), and, as we believe, occurs (though broadened by temperature) in liquids such as fluid caesium (Chapter 10). The disorder here is greater than in a liquid metal because the orbitals of the electrons in the donors can overlap strongly. The present author (Mott 1978) has given conditions under which disorder can remove the discontinuity but this may not be relevant to such materials as Si P, because (Section 12) the Hubbard gap has disappeared, at any rate in many-valley semiconductors, at a concentration well below the transition,... 

Anderson type (though affected of course by long-range interaction). Until recently it was supposed by the present author that the former is the case. We must now favour, however, the latter assumption for many-valley materials (e.g. Si and Ge), the Hubbard gap opening up only for a value of the concentration n below nc. The first piece of evidence comes from a calculation of Bhatt and Rice (1981), who found that for many-valley materials this must be so. The second comes from the observations of Hirsch and Holcomb (1987) that compensation in Si P leads to localization for a smaller value of nc than in its absence. As pointed out by Mott (1988), a Mott transition occurs when B = U (B is the bandwidth, U the Hubbard intra-atomic interaction), while an Anderson transition should be found when B 2 V, where V is some disorder parameter. Since U e2/jcuH, where aH is the hydrogen radius, and K e2/jca, and since at the transition a 4aH, if the transition were of Mott type then it should be the other way round. 

The addition of 2.5% Cr02 leads at intermediate temperature to a phase i n which only half the V ions are paired the others form a zig-zag chain (Marezio et al 1972, Pouget et al 1974). At low temperatures pairing takes place, and at higher temperatures the usual transition to the metallic rutile form. This intermediate phase has high susceptibility, and the zig-zag chains are interpreted as onedimensional Mott-Hubbard insulators above their Neel temperature. Since the transition temperature is little changed, this shows that U is the most important quantity in determining the gap. 

Mott-Hubbard metal-insulator transition in the nanocrystal ensemble wherein the Coulomb gap closes at a critical distance between the particles. 

We may incidentally remark that the same kinds of arguments could also be reasonably invoked for the electrical properties of some materials of more recent concern. For instance, a resistivity minimum that is not related to a phrase transition is observed at Tp in (DMDCNQI)2Ag (DMDCNQI = dimethylcicyanoquinonediimine) [59] as in (TMTTF)2PF6 [60]. This minimum is attributed to the opening of a gap in the electronic spectrum of the materials below Tp, under the effect of Umklapp scattering, with a corresponding Mott-Hubbard localisation [14]. It is, however, also... 

Figure 7.1. The band gap is determined by the d-d electron correlation in the Mott-Hubbard insulator (a), where A > I/. By contrast, the band gap is determined by the charge transfer excitation energy in the charge transfer insulator (b), where U > A.

Unlike the Mott-Hubbard insulator MnO described above the band-gap in the isostruc-tural oxide NiO is much smaller than expected from intrasite Coulomb repulsion. Fujimori and Minami showed that this is owing to the location of the NiO oxygen 2p band - between the lower and upper Hubbard sub-bands (Fujimori and Minami, 1984). This occurrence can be rationalized by considering the energy level of the d band while moving from Sc to Zn in the hrst transition series. 

In addition to Mott-Hubbard localization, there is another common source of electron localization, which arises when a lattice is under a random potential (e.g. a random distribution of alkali metal ions in alkali metal containing transition metal oxides). For a metal, a practical consequence of a random potential is to open a band gap at the Fermi level. Insulating states induced by random potentials are referred to as Anderson localized states (see Anderson Localization)) ... 

The ARPES spectra signal for the sulfur compound (TMTTF)2PF,5, displays a rigid shift of the leading edge near the Fermi energy to about 100 meV. This value is consistent with the charge gap of 900 K obtained from transport experiments DC or 800 cm from optical conductivity of (TMTTF)2PF6, Within a onedimensional frame of interpretation, this gap has been ascribed to a Mott-Hubbard localization gap [99],... 

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