Band crossing

For the nanotubes, then, the appropriate symmetries for an allowed band crossing are only present for the serpentine ([ , ]) and the sawtooth ([ ,0]) conformations, which will both have C point group symmetries that will allow band crossings, and with rotation groups generated by the operations equivalent by conformal mapping to the lattice translations Rj -t- R2 and Ri, respectively. However, examination of the graphene model shows that only the serpentine nanotubes will have states of the correct symmetry (i.e., different parities under the reflection operation) at the K point where the bands can cross. Consider the K point at (K — K2)/3. The serpentine case always sat-... 

Fig. 6. Examples of (a) mirror symmetry and (b) non-symmetry with respect to the tube axis. The HO-LU band crossing in (a) changes into avoid crossing in (b). Notations S and A signify symmetric and antisymmetric with respect to the mirror symmetry, respectively, for instance.

Appearance of the metallic structure of CNT is based on the crossing of the highest occupied (HO) and the lowest unoccupied (LU) bands (see, e.g.. Fig. 3), each accompanying pseudo rt-type crystal orbital. Note that pseudo n-type orbital, particularly when all the valence atomic orbitals (AO) are taken into consideration, implies that its main AO component is normal to the cylindrical CNT surface. The band crossing mentioned above is possible when these two... 

FIGURE 17.12 Electron micrograph of a skeletal muscle myofibril (in longitndinal section). The length of one sarcomere is indicated, as are the A and I bands, the H zone, the M disk, and the Z lines. Cross-sections from the H zone show a hexagonal array of thick filaments, whereas the I band cross-section shows a hexagonal array of thin filaments. (Photo courtesy of Hugh Huxley, Brandeis University)... 

Copper is a face-centered cubic (fee) metal. Band structure calculations show the valence bands to be copper d bands and hybrid bands of sd, pd, and sp character. The hybridization is essential for the conductivity of copper, as some of the bands cross the Fermi surface and are thus only partially occupied (K. Schwarz, private communication). 

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...

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

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. 

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. 

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