Band-crossing transitions

This is a much smaller energy than that of a free exdton (—m e4/2h2K2), and a condensed gas of exdtons is a possibility that has been discussed in the literature 

The application to metal-insulator transitions is as follows. As the quantity AE in Fig. 4.1 decreases, and before it disappears but when AE=Ecrit, the number of carriers will, at zero temperature, increase from zero to a value of nc given by (2). If k is large, as it normally will be because the band gap is small, the discontinuity may be small and observable only at low temperatures. 

In principle, then, the zero-temperature free energy of the system, plotted against volume or (in an alloy) composition, both denoted by x, must show a kink as illustrated in Fig. 4.2 at the metal-insulator transition. If x is the volume and this is decreased by pressure then there will be a discontinuous change of volume between B and A. If x denotes the composition then between B and A the alloy will be unstable, and will in equilibrium separate into two phases. The behaviour 

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These effects have not been observed in band-crossing transitions, but have in the so-called Mott-Hubbard transitions described in the next section. 

The metal-insulator transition may perhaps be envisaged as similar to a band-crossing transition, caused by overlap between the Cf states. Since these are negatively charged, if the wave function is expressed as (1) (2)+W2) fc(l), where ij/a and jfb are both s-functions with different radii, then the larger radius could be considerable. We think that this may account for the small value of <7mill observed (cf. Section 4), the number of free electrons at the transition being small. 

An early success of quantum mechanics was the explanation by Wilson (1931a, b) of the reason for the sharp distinction between metals and non-metals. In crystalline materials the energies of the electron states lie in bands a non-metal is a material in which all bands are full or empty, while in a metal one or more bands are only partly full. This distinction has stood the test of time the Fermi energy of a metal, separating occupied from unoccupied states, and the Fermi surface separating them in k-space are not only features of a simple model in which electrons do not interact with one another, but have proved to be physical quantities that can be measured. Any metal-insulator transition in a crystalline material, at any rate at zero temperature, must be a transition from a situation in which bands overlap to a situation when they do not Band-crossing metal-insulator transitions, such as that of barium under pressure, are described in this book. 

In transitional and noble metals the s-band crosses the d-band and is hybridized with it. The situation is discussed in a number of papers (cf. Mott 1964, Heine 1969, p. 25). Figure 1.7, taken from Heine (1969), shows the band structure of copper in the (111) direction. 2y is the hybridization gap where the 4s-band crosses the d-band. ... 

Fig. 4.1 Metal-insulator transition of band-crossing type. E(k) is plotted against k in the...

Anderson versus band-crossing or Mott transitions... 

It would be of interest to have measurements at lower temperatures to see if metal-insulator transition of band-crossing type occurs, the... 

A prediction of theory (Chapter 4) is that when an insulator-metal transition of either band-crossing or Mott type occurs through a change of composition in an alloy, the zero-temperature conductivity should jump discontinuously from zero to a finite value. This seems to be the case for the alloys with Ti203. The alloys with titanium have a conductivity when metallic of order 104 1 cm-1 at... 

Mott (1980 a) has given a further argument against the assumption that two electrons in the same cavity form the lowest state of the system. If this were so then the transition would be of band-crossing type, and one would not expect the observed two-phase region. 

Band crossings and other features of the mass 100 transitional... 

An interesting case of the electroplex formation may be expected for the low ionization potential of electron donor and high electron affinity of electron acceptor molecules. The electron transition would then occur at large intermole-cular distances (Ec kT) and the optical cross-transition appears at hucross=Ij)-AA [cf. Eq. (274) with AE = 0]. The broad band at =550 nm in the EL spectrum of a DL LED... 

Tlie bands have long been interpreted as LCAO d bands, crossed by and hybridized with a free-electron-like band (Saffren, 1960 Hodges and Ehrenreich, 1965 and Mueller, 1967). By including some eleven parameters (pseudopotential matrix elements, interatomic matrix elements, orthogonality corrections, and hybridization parameters), it is possible to reproduce the known bands very accurately. We shall also make an LCAO analysis of the bands but shall take advantage of recent theoretical developments to reduce the number of independent parameters to two for each metal, each of which can be obtained for any metal from the Solid State Table. These two parameters will also provide the basis for understanding a variety of properties of the transition metals. 

The electronic bands of an infinite crystal can cross as a function of some parameter (pressure, concentration etc.). If one treats the e /r,2 term of the electron repulsion correctly, one sees that the crossing transition of the two bands is a first-order phase transition, between the metallic and insulating states. This transition was predicted by Mott in 1946 and has carried his name ever since. In fact, the original Mott criterion does not predict such a transition for Hg, but the criterion was derived for monovalent atoms. For divalent mercury it should not be applicable. Also the semiempirical Herzfeld criterion, which was very successful in predicting the insulator to metal transition in compressed xenon, predicts bulk Hg to be non-metallic. All this seems to imply that Hg is a rather special case. 

Band crossing of the Fermi level is also supposed to be the essential physics of static quantum size effects (SQSE) in metal UTFs. Here the control parameter is UTF thickness rather than compression as in the transition metal ILs just discussed. JCB and SBT have published extensively on this problem, so we only summarize a few of the most interesting features here. Jellium models [104] predict thickness-dependent oscillations in the wotk-fiinction caused by new bands touching the Fermi level as thickness is increased. Our calculations [90] on Li... 

Mercury, values of properties. See Simple metals Metal-induced surface states, 246, 428 Metal-insulator transition, 39, 42 by band crossing, 39, 163 transition-metal monoxides, 437f Metal-semiconductor junctions, 244fT, 428, 429pr Metals. See Simple metals Transition metals Metallic energy, 67 perovskites, 457ff table, 50... 

For this latter reason, for example, it was not possible to use spectroscopy to evaluate the various equilibria in the hydrolysis of uranyl ion in perchlorate media. Without assignment of all bands to their appropriate transitions it was not possible to rule out band crossing or overlapping, with pH change. 

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