Hubbard bands

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. 

The antiferromagnetic state described by the occupation of the lower Hubbard band is stabilized by inclusion of such electron correlation, but the ferromagnetic analog is not. This is a result exactly analogous to the stabilization of the lowest singlet state in cyclobutadiene below the triplet. For the simple density of states used by Hubbard in his treatment he showed in fact that the condition for ferromagnetism was... 

Figure 15 (a) One-band model where electrons are removed from the lower Hubbard band on doping, (b) One-band model where electrons are removed from the tr band of the material on doping, (c) Similar to (b) but where the system is a normal metal described by (W/U) > (W/U)crit of Figure 11. 

Electron correlation plays an important role in determining the electronic structures of many solids. Hubbard (1963) treated the correlation problem in terms of the parameter, U. Figure 6.2 shows how U varies with the band-width W, resulting in the overlap of the upper and lower Hubbard states (or in the disappearance of the band gap). In NiO, there is a splitting between the upper and lower Hubbard bands since IV relative values of U and W determine the electronic structure of transition-metal compounds. Unfortunately, it is difficult to obtain reliable values of U. The Hubbard model takes into account only the d orbitals of the transition metal (single band model). One has to include the mixing of the oxygen p and metal d orbitals in a more realistic treatment. It would also be necessary to take into account the presence of mixed-valence of a metal (e.g. Cu ", Cu ). 

For a transition of Mott type we shall show in Chapter 4, Section 3, neglecting the discontinuity resulting from long-range forces, that the transition should occur when 2zl = U. Near the transition the energy needed to excite an electron into the upper Hubbard band is U — 2zl. The wave function of an electron then falls off as e-fltr, where a=2m(I7 — 2zI)1/2/fc2. Thus the amount of spin in the sphere surrounding each atom will be made up from electrons on many of the surrounding atoms, and will clearly go to zero as [Pg.88]

Here we have in mind such materials as EuS with a comparatively high concentration of Gd atoms to give a degenerate electron gas, and a large number of metallic transitional-metal compounds where ions of mixed valence exist (in the latter there may be uncertainty about whether the electrons are in a conduction (4s) band or the upper Hubbard band described in Chapter 4). In such a case a new interaction term arises between the moments which is via the conduction electrons. This is the so-called RKKY (Ruderman-Kittel-Kasuya-Yosida) interaction, which is an oscillating function of distance (Ruderman and Kittel 1954, Kasuya 1956, Yosida 1957 for a detailed description see Elliott 1965). This derives from the formulae of Chapter 1, Section 5. Consider an atom with magnetic moment in a given direction then the wave functions of conduction electrons with spin up and with spin down will vary with distance in different ways, so that... 

Our model is thus of a metal, with a small number of carriers in the conduction band and 3d- or 4f-moments antiferromagnetically coupled to each other by direct exchange. The carriers may be either inserted by doping or by overlap from the lower Hubbard band (cf. Chapter 4, Section 3). The most striking prediction of the model, however, is that the degenerate electron gas should have a much enhanced Pauli magnetism. Suppose that EF is the Fermi... 

Fig. 4.4 The motion of (a) an electron in an upper Hubbard band and (b) a hole, marked...

Even when B U, this is still not large, and if jumps to nearest neighbours were allowed only on account of this term then the mass of a carrier at the extremities of the Hubbard bands would be much enhanced. We think, however, that a more important effect allowing the carrier to move to nearest neighbours is the formation of spin polarons. These will be considered in Section 4. 

Such spin polarons should not have a mass much greater than m in Si P. Moreover, they can pass freely from one atom to another, and are not impeded by the antiferromagnetic order. Thus the bandwidth of each Hubbard band should, we believe, still be of order 2zl, as it is for large values of U/B, and the equation... 

If the spin-polar on model is correct, we must describe the carriers in the antiferromagnetic semimetal formed when the two Hubbard bands overlap as a degenerate gas of spin polarons it should have the following properties. 

Ramirez et al (1970) discussed a metal-insulator transition as the temperature rises, which is first order with no crystal distortion. The essence of the model is—in our terminology—that a lower Hubbard band (or localized states) lies just below a conduction band. Then, as electrons are excited into the conduction band, their coupling with the moments lowers the Neel temperature. Thus the disordering of the spins with consequent increase of entropy is accelerated. Ramirez et al showed that a first-order transition to a degenerate gas in the conduction band, together with disordering of the moments, is possible. The entropy comes from the random direction of the moments, and the random positions of such atoms as have lost an electron. The results of Menth et al (1969) on the conductivity of SmB6 are discussed in these terms. 

However, we do not think that this model is applicable to V203 and its alloys, chiefly because we do not think a conduction band necessarily lies near the lower Hubbard band, and it does not describe many important features, particularly the large mass enhancement (approximately 50) in the metallic phase. In our view, to understand the behaviour of such materials as the temperature is raised, and in... 

Evidence that the transition ties in an impurity band, and that the two Hubbard bands have merged... 

Fig. 5.13 Schematic illustration of how two Hubbard bands, with localized tails (shaded), resulting from disorder, can overlap, so that the equation (Bl+B2) U determines approximately the concentration at which the transition occurs, while the properties of the materials near the transition are those resulting from a transition of Anderson type.

In earlier work (see e.g. Mott 1987) the present author has attempted to combine the hypothesis that the Hubbard U determines the value of nc in doped semiconductors with the observation that the transition shows the properties of one of Anderson type (second order, cv = 0, quantum interference and interaction effects) by supposing that two Hubbard bands, separated by U, have small localized tails, as in Fig. 5.13, and that the transition occurs for a value of nc such... 

Whether the carriers remain within the eg band is uncertain. Honig considers that excitation from the 3d- to the 4s-band may be a possibility, the upper Hubbard band for the eg states lying at a higher energy. In this case no polarons would be expected for the electrons, but would for the holes. Thus we should expect Es=E[Pg.172]

The low-temperature phase under pressure shows a fairly high conductivity, as one would expect from increasing overlap of two Hubbard bands. 

In NaxW03-yFy Doumerc (1978) observed a transition that has all the characteristics of an Anderson transition similar phenomena are observed in NaxTayW3 y03. The results are shown in Fig. 7.14. It is unlikely that this transition is generated by the overlap of two Hubbard bands with tails (Chapter 1, Section 4) this could only occur if it took place in an uncompensated alkali-metal impurity band, which seems inconsistent with the comparatively small electron mass. We think rather that in the tungsten (or tungsten-tantalum) 5d-band an Anderson transition caused by the random positions of Na (and F or Ta) atoms occurs. The apparent occurrence of amiD must, as explained elsewhere, indicate that a at the temperature of the experiments. Work below 100 K, to look for quantum interference effects, does not seem to have been carried out. 

From 3 to (say) 0.3 MPM the conductivity is not in the metallic range, but we believe that the material behaves like an intrinsic semiconductor, current being carried by electrons in the upper band. This, as we have seen, is not a Hubbard band, but the band formed from molecular orbitals for an extra electron on molecular dimers. In this range d In a jdT... 

A further assumption is that the disorder produces tails to the two Hubbard bands, which overlap, and that the transition occurs when states become delocalized at F, as described in Chapter 5, Section 12. 

To study this regime, a combination of large-17 and small-17 orbitals is considered, where major aspects within the CuC>2 planes are approached by the t—t —J model. The large-17 electrons, residing in these planes, are treated by the slave-fermion method [1], Such an electron in site i and spin a is created by d a =, ifit is in the upper-Hubbard-band , and by = cxsjahi, ifit... 

When the cuprates are doped, such QE states are transferred from the upper and lower Hubbard bands to the vicinity of the Fermi level (ftp). The amount of states transferred is increasing with the doping level, moving from the insulating to the metallic side of the Mott transition regime. 

Figure 7 Sketch of the processes responsible for (a) spin or (b) charge excitations in a half-filled Hubbard band.

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