Metal-Insulator Transitions in Crystals

The present author (Mott 1949,1956,1961) first proposed that a crystalline array of one-electron atoms at the absolute zero of temperature should show a sharp transition from metallic to non-metallic behaviour as the distance between the atoms was varied. The method used, described in the Introduction, is now only of historical interest. Nearer to present ideas was the prediction (Knox 1963) that when a conduction and valence band in a semiconductor are caused to overlap by a change in composition or specific volume, a discontinuous change in the number of current carriers is to be expected a very small number of free electrons and holes is not possible, because they would form exdtons. 

The next step in our understanding of the transition for one-electron centres was the work of Hubbard (1964a, b), who introduced the Hamiltonian 

Here U is the intra-atomic interaction e2/r12 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. 

In this book we treat the discontinuous nature of the transition using an analysis introduced by Brinkman and Rice (1970a, b). This applies to bandcrossing transitions and transitions in an array of one-electron centres. We term the latter Mott transitions when the centres have a moment we do not limit the term to cases when the moment is that of a single spin, and indeed such cases are rare (Chapter 3). The insulating antiferromagnetic state is sometimes called a Mott insulator . A Mott transition can be accompanied by a change of structure (see Section 3 below). 

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Figure 10. The size-induced metal-insulator transition in mesoscopic crystals of indium. The size-dependent [quasi) d.c. conductivity versus particle diameter. For comparison the bulk conductivity and the classical (surface) size-effect are also displayed. Modified from Nimtz et alP° ...

The phase transition described above can be characterized as a second order transition. However, the nature of the metal-insulator-transition in FA2X-salts is still an unsolved question a a matter of continuing research. In DSC experiments two transitions at approximately 200 K an 180 K are observed of which the first one is the transition which can be monitored crystallographically. However, the electronic properties change abruptly at 180 K. The assumptic of the existence of a lower transition is also supported by some results of ESR- and NMR-experiments with a pronuonced line broadening at and below the phase transition around K as well as by measurements of the static susceptibility Wd the electrical conductivity. At thi temperature no anomalies in the temperature dependence of the crystal structures can be detected... 

Anderson localization is only one possible way of inducing a transition from a localized to a delocalized state. Since the localized state is associated with insulating behavior and the delocalized state is associated with metallic behavior, this transition is also referred to as the metal-insulator transition. In the case of Anderson localization this is purely a consequence of disorder. The metal-insulator transition can be observed in other physical situations as well, but is due to more subtle many-body effects, such as correlations between electrons depending on the precise nature of the transition, these situations are referred to as the Mott transition or the Wigner crystallization. 

In situations where the performance is dependent on the uniqueness of the crystal structure of the material, such as the metal-Insulator transition in vanadium oxide, acousto-optic and acoustic-electric responses of AIN, ZnO, PZT, BaTiOg and SrTi03, superconducting properties of copper oxide based perovskites, A-15 silicides, and NbN, wide band gap and dopability of SiC, transistor action in Si/NiSi2(CoSi2)/Sl epitaxial layers etc., it is important to optimize the deposition conditions for the growth of a %[Pg.395]

In the framework of CUORICINO [41] and CUORE [42] experiments (see Section 16.5), Ge crystal wafers of natural isotopic composition have been doped by neutron irradiation, and the heavy doping led to materials close to the metal insulator transition. Several series of NTD wafers with different doping have been produced [43], After an implantation and metallization process on both sides of the wafers, thermistors of different sizes can be obtained by cutting the wafers and providing electrical contacts. 

Titanium dioxide crystallizes in several forms. The most important is the rutile form. This structure is also adopted by S11O2, MgF2, and ZnF2. A number of oxides that show metallic or metal-insulator transitions, for example, VO2, NbC>2, and Cr02, have a slightly distorted form of the structure. 

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). 

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. 

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. 

Block-shaped black crystals with a distinct metallic luster are formed which upon crystallographic study are found to crystallize in the triclinic system with lattice constants as follows a = 7.786 (1) A, fe = 13.033 (3) A, c = 18.590 (4) A, a = 110.09 (2)°, 0 = 90.21 (2)°, y = 105.27 (2)°, and Vc = 1699.8 (6) A3. These lattice parameters indicate that the crystals contain 0.5 molecules of TCE in the formula unit as is the case for isostructural (BEDT-TTF)2ClO4(TCE)0 5 in which Vc = 1689.8 A3.10 The electrical properties of this material indicate that they are metallic from room temperature to 90 K at which point a metal-insulator transition occurs. 

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