A metal-insulator transition

On Fig. 3 we plot the probabilities of different configurations versus the direct Coulomb interaction U. It can be seen that the system undergoes a metal-insulator transition for a sufficiently high value of U, close to 9. It is easy to perform the same kind of calculation in the case of triply degenerate... 

The trisulphides (and triselenides) of Ti, Zr, Hf, Nb and Ta crystallize in onedimensional structures formed by MSg trigonal prisms that share opposite faces. Metal atoms in these sulphides are formally in the quadrivalent state, and part of the sulphur exists as molecular anions, M S2 S . TaSj shows a metal-insulator transition of the Peierls type at low temperatures (Section 4.9). NbSj adopts a Peierls distorted insulating structure suggesting the possibility of a transformation to a metallic phase at high temperatures, but does not transform completely to the undistorted structure. Electronic properties and structural transitions of these sulphides have been reviewed (Rouxel et al, 1982 Meerschaut, 1982 Rouxel, 1992). 

The present author (Mott 1949, 1956, 1961) described a metal-insulator transition by imagining a crystalline array of hydrogen-like atoms with a lattice constant a that could be varied. The example of nickel oxide suggested that for large values of a the material would be insulating, while the example of monovalent metals like sodium showed that for small values it would be metallic. 

The question was, at what value of a would a metal-insulator transition occur The assumption (Mott 1949) was made that this would occur when the screened potential round each positive charge,... 

Kirk et al (1972) show that at 6.5 kbar there is a discontinuous change in the reflectivity of SmS, suggesting a metal-insulator transition, which was later observed by Bader et al (1973) with an 8% volume change (see also Chapter 4 below and Jayaraman et al 1970a, b, Maple and Wohlleben 1971). 

An example of a very small metallic conductivity (Honig 1985) is quoted in Chapter 6 in our discussion of (V xCrJ203 just oh the metallic side of a metal-insulator transition as the temperature is increased. A value of a, little dependent... 

With neglect of multiple scattering, Mott (1972a) supposed that if EF—EC changed sign, causing a metal-insulator transition, or if a transition at mid-band resulted from a change of V then the metallic conductivity would drop to a minimum value... 

For the subject matter of this book, it is of particular interest to consider the situation for a non-crystalline system analogous to that of crystalline ytterbium or strontium under pressure, namely that when a valence and conduction band are separate or overlap slightly. If the degree of overlap can be changed by varying the mean distance between atoms, the composition or the coordination number then a metal-insulator transition can occur. Many examples will be discussed in this book, particularly amorphous films of composition (Mgi- )j(By3, liquid mercury at low densities, and liquid tellurium alloys in which the coordination number changes with temperature. The transition is, we believe, of Anderson type. 

We emphasize that the use of g in these equations may be justified only if /—a, because of the Edwards cancellation theorem (Section 6). We should expect a metal-insulator transition to occur for some value of in the neighbourhood of For several liquid systems there is experimental evidence that the interference term in (52) is absent. Thus for liquid TeTl alloys, with variation of composition and temperature, for a less than the Ioffe-Regel value e2/3hai the conductivity is proportional to the square of the Pauli paramagnetic susceptibility and then to 2. These results are due to Cutler (1977). Warren (1970a, b, 1972a, b) examined... 

We discuss in this section the effect of short-range interaction on the Anderson-localized states of a Fermi glass described in Chapter 1, Section 7, and in particular the question of whether the states are singly or doubly occupied. Ball (1971) was the first to discuss this problem. In this section we consider an electron gas that is far on the metal side of the Wigner transition (Chapter 8) the opposite situation is described in Chapter 6, where correlation gives rise to a metal-insulator transition. We also suppose that Anderson localization is weak (cca 1) otherwise it is probable that all states are singly occupied. 

Here U is the intra-atomic interaction defined in Chapter 4 and t, the hopping integral, is equal to B/2z, where B is the bandwidth and z is the coordination number. The suffixes i and j refer to the nearest-neighbour sites, and aia is the creation operator for site i. The suffix a refers to the spin direction. Hubbard found that a metal-insulator transition should occur when B/U = 1.15. Hubbard s analysis did not include long-range interactions, and therefore did not predict any discontinuity in the number of current carriers. 

If we neglect the electron-hole interaction, as in Chapter 1, then a metal-insulator transition should occur when the two bands overlap. For infinite values of a, the separation in energy between the two bands should be just the Hubbard Uy so the transition occurs when... 

In materials in which a metal-insulator transition takes place the antiferromagnetic insulating state is not the only non-metallic one possible. Thus in V02 and its alloys, which in the metallic state have the rutile structure, at low temperatures the vanadium atoms form pairs along the c-axis and the moments disappear. This gives the possibility of describing the low-temperature phase by normal band theory, but this would certainly be a bad approximation the Hubbard U is still the major term in determining the band gap. One ought to describe each pair by a London-Heitler type of wave function... 

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. 

Impurity conduction can also be studied in compensated semiconductors, i.e. materials containing acceptors as well as donors, the majority carriers (or the other way round). For such materials, even at low concentrations, activated hopping conduction can occur (Chapter 1, Section 15), some of the donors being unoccupied so that an electron can move from an occupied to an empty centre. Here too a metal-insulator transition can be observed, which is certainly of Anderson type, the insulating state being essentially a result of disorder. 

In Si P and some other systems in which a metal-insulator transition takes place, such as the liquids K-KC1 (see e.g. Warren 1985) and NaxNH3 (Chapter 10) the dielectric constant can become very large as the transition is approached from below, for example in Si P by increasing the phosphorus content. Some... 

In NiO a metal-insulator transition has been observed under very high pressure (2.5Mbar) (Kawai and Mochizuki 1971) the conductivity at room temperature dropped abruptly by about 106. Nothing was determined regarding the temperature dependence of the conductivity, or the change of volume. 

NiS2, particularly its behaviour under pressure, has been discussed by Wilson and Pitt (1971) and Wilson (1985). This has two electrons half Ming an eg band, and should therefore behave like V203. It is an antiferromagnetic semiconductor, but shows a metal-insulator transition at a pressure of 46kbar (Mori et al 1973). Thus the transition occurs for a decrease in volume of about 0.4%, with no change... 

Figure 7.6 shows the resistivity plotted against 1/Tfor various values of x. A metal-insulator transition occurs with x 0.8 and ffmin=2x 103ft 1 cm-1, which is rather a large value—perhaps because the orbitals are d-like (see... 

Metal-ammonia solutions show a dramatic increase in the dielectric constant as the concentration approaches that for a metal-insulator transition from below. We think this has to be interpreted as in Chapter 5, Section 8. Investigations of this behaviour through ESR measurements in LixNH3 have been made by Damay et al (1988), with a theoretical interpretation given by Leclercq and Damay (1988). 

Most amorphous metallic alloys do not show a metal-insulator transition. They do, however, show moderate changes in the resistivity with temperature, some of which can be interpreted in terms of the quantum interference effect, together with the interaction effect of Altshuler and Aronov (Chapter 5, Section 6). These will be described below. Amorphous alloys of the form Nb Six Au Six etc. do, however, show a metal-insulator transition of Anderson type, and some of those are treated in Chapter 1, Section 7. 

Crystals of (BEDT-TTF)2Re04 are lustrous metallic black in color and are superconducting at a temperature of 2 K at a pressure of 5 k bar.5 In the absence of applied pressure this material shows metallic behavior (increased electrical conductivity as the temperature is decreased) from room temperature to 81 K at which temperature a metal-insulator transition occurs. The crystallographic lattice parameters, for the triclinic unit cell, are5 a = 7.78 A, b = 12.59 A, c — 16.97 A, a = 73.0°, 0 = 79.89°, y = 89.06°, and unit cell volume, Vc = 1565 A3. 

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