Van Hove singularity

Clark et al.n recently discovered another FS related inechanisni in CuPt, different from the above mentioned nesting. In this case, the relevant contribution to the coneentration waves suseeptibility is due to the contemporary presenee of the noble metal-like neck at the L point and the d hole pocket at X. Ifie connecting vector of these Van Hove singularities belongs to the star 1,1,1 and is commensurate with the Lli ordering. In fact, it produces a phase characterised by alt ate hexagonal Cu and Pt planes, in the direction perpendicular to (1,1,1). 

A dramatic hybridization splitting around the crossing between the dispersionless adlayer mode and the substrate Rayleigh wave (and a less dramatic one around the crossing with the co = CiQg line - due to the Van Hove singularity in the projected bulk phonon density of states). 

A final observation from Fig. 8.3 is that there are a number of places where the slope of the DOS changes discontinuous. These are known as van Hove singularities. 

We see that the structural trend from fee - bcc hep is driven by the van Hove singularities in the densities of states. These arise whenever the band structure has zero slope as occurs at the bottom or top of the energy gaps at the Brillouin zone boundaries. The van Hove singularities at the bottom of the band gap at X and at the top of the band gap at L in fee copper are marked X4. and Ly, respectively, in the middle panel of Fig. 6.16. It is, thus, not totally surprising that the reciprocal-space representation... 

Therefore, as shown in Fig. 7.1(a), the bottom of the band is at the centre of the Brillouin zone (0,0,0), whereas the top of the band is at the zone boundary ( / )( 1,1,1), since ssa < 0. It follows from eqn (7.1) that the bottom and top of the band correspond to perfect bonding and antibonding states, respectively, between all six neighbouring atoms, so that the width of the s band is 2 6ssoj, as expected. The corresponding density of states is shown in Fig. 7.1(b). The van Hove singularities, arising from the flat bands at the Brillouin zone boundaries, are clearly visible. 

If (n — m( = 3q, where q is an integer, the SWNT is metallic, whereas for (n — m ( 3q, it is semiconducting with a bandgap in the density of states (DOS) whose size is inversely proportional to the diameter. As a consequence of the size-dependent quantization of electronic wave functions around the circumference of the SWNT, the DOS shows typical singularities, the so-called van Hove singularities, consisting of a... 

About one-third of all the SWNTs are metallic and always have wider energy gaps between the first van Hove spikes than semiconducting ones with similar diameter. The presence of the van Hove singularities dominates the spectral features of these species58 as well as the electrochemical ones.59... 

Figure 1 2 1. The different types of 2.5 Lifshitz electronic topological transition (ETT) The upper panel shows the type (I) ETT where the chemical potential EF is tuned to a Van Hove singularity (vHs) at the bottom (or at the top) of a second band with the appearance (or disappearance) of a new detached Fermi surface region. The lower panel shows the type (II) ETT with the disruption (or formation) of a neck in a second Fermi surface where the chemical potential EF is tuned at a vHs associated with the gradual transformation of the second Fermi surface from a two-dimensional (2D) cylinder to a closed surface with three dimensional (3D) topology characteristics of a superlattice of metallic layers...

The interband pairing term enhances Tc [93-97,102] by tuning the chemical potential in an energy window around the Van Hove singularities, z =0, associated with a change of the topology of the Fermi surface from ID to 2D (or 2D to 3D) of one of the subbands of the superlattice in the clean limit. 

Optical density spectra of thin films of the initial SWNTs also demonstrate distinct absorption peaks at 0.62,1.13 and 1.65 eV (spectrum 1 of Fig. 11.7). These absorption lines result from the van Hove singularities in the density of electron states due to the one-dimensional nature of nanotubes (Lin and Shung 1994). 

The first integral, which has the largest integration range, is easy to calculate even though there are many van Hove singularities in the integrand. 

One of the two integrals (4.67) may be evaluated analytically the second one (elliptic) may be calculated numerically by the trapezoidal method with the introduction of a small imaginary part in (4.67) to avoid spurious oscillations due to the finite number of integration points. The resulting density-of-states function is shown in Fig. 4.9. The band is asymmetric because of the equivalent term Vx j it exhibits three van Hove singularity points two discontinuities at the boundaries, and one logarithmic divergence corres-... 

The second symmetry is the consequence of the local symmetry of the density of the states of the resonant band. Finite, small t modifies strongly the dispersion of the conducting band in the vicinity of the van Hove singularities, adding extra states below the Fermi level. The Fermi energy at the half-filling is therefore shifted from the van Hove singularity towards the (n, w) point when t < 0. 

Thus the band-properties have the symmetry with respect to the doping 5C [8] required to bring the Fermi energy back to the van Hove singularity, as observed in the transport data. 

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