Electronic dispersion curve

Since the superconductive phase contains itinerant charge carriers and a well-defined Fermi surface, interpretation of the enhancement of a(T) must begin with the general expression of Eq. (5). It follows from Eq. (5) that the enhancement implies an increasing asymmetry of the electronic dispersion curve on crossing Sp and therefore an increased flattening of the dispersion... 

Iditional importance is that the vibrational modes are dependent upon the reciprocal e vector k. As with calculations of the electronic structure of periodic lattices these cal-ions are usually performed by selecting a suitable set of points from within the Brillouin. For periodic solids it is necessary to take this periodicity into account the effect on the id-derivative matrix is that each element x] needs to be multiplied by the phase factor k-r y). A phonon dispersion curve indicates how the phonon frequencies vary over tlie luin zone, an example being shown in Figure 5.37. The phonon density of states is ariation in the number of frequencies as a function of frequency. A purely transverse ition is one where the displacement of the atoms is perpendicular to the direction of on of the wave in a pmely longitudinal vibration tlie atomic displacements are in the ition of the wave motion. Such motions can be observed in simple systems (e.g. those contain just one or two atoms per unit cell) but for general three-dimensional lattices of the vibrations are a mixture of transverse and longitudinal motions, the exceptions... 

Detailed electronic energy-band calculations have revealed the existence of appropriate surface states near the Fermi energy, indicative of an electronically driven surface instability. Angle-resolved photoemission studies, however, showed that the Fermi surface is very curved and the nesting is far from perfect. Recently Wang and Weber have calculated the surface phonon dispersion curve of the unreconstructed clean W(100) surface based on the first principles energy-band calculations of Mattheis and Hamann. ... 

This mechanism is identical to the one arising from the contact interaction between an unpaired electron and a nuclear spin (41). In that case, the hyperfine coupling (generally denoted by Asc or A and exists only if the electron density is non-zero at the considered nucleus, hence the terminology of contact ) replaces the J coupling and the earlier statement (i) may be untrue because it so happens that T becomes very short. In that case, dispersion curves provide some information about electronic relaxation. These points are discussed in detail in Section II.B of Chapter 2 and I.A.l of Chapter 3. 

There is a condition of momentum conservation for photons and electrons which must also be satisfied in the photoemission process. For band electrons, for which the Bloch wavefunctions are characterized by the wavenumber k (proportional to the momentum p of the electron), the momentum conservation condition is important to determine the angular distribution of the photoemitted electrons. Angular J esolved FhotoEmission spectroscopy (ARPES), schematized in Fig. 2, is potentially able to provide, and has been used to obtain, the E(fc) dispersion curves for solids. 

What is the form of the general dispersion curve describing the momentum-energy relationship for a quasi-free electron in a liquid ... 

For high electron densities n, scattering at the pure plasmon is obtained, whereas for relatively low values of n coupled plasmon-phonon states are observed. In Fig. 8 the dispersion curves for gallium arsenide measured by Mooradian et al. 

The conductance for a spacing of 2 /xm between gates g and g2 is shown in Fig. 2. The measured bright and dark curves in the plot can be interpreted as spectral peaks tracing out the dispersions of the elementary excitations in the wires. [3] In the case of noninteracting electrons, the curves are expected to map out parabolas defining the continua of electron-hole excitations across... 

Starting from this simple yet physically robust model of the band structure, the discussion of the electronic structure can be extended to include the effect of the Coulomb interaction. The delocalized nature of Dl and Dl implies that, for electron correlation effects involving these subbands, it is appropriate to compute an effective mass from the dispersion curves and then compute the corresponding one-dimensional (Id) hydrogenic levels. From the k -dependence of the dispersion near the zone center, the effective masses in Dl and Dl are m = 0.067 m,. 

The n-type conductivity is determined by the product of the reciprocal effective mass of electrons and the concentration of carriers in a semi-classical viewpoint. The effective mass is often calculated by fitting the dispersion curve near the bottom of the conduction band. The bottom is mainly composed of M-4s/5s orbitals in the oxides of the present interest, and the curvature is mainly determined by the M-M interactions. The situation is schematically shown in Fig. 4. The strong interaction among M-4s/5s orbitals should bring about wider M-4s/5s band-width and smaller effective mass. On the other hand, weaker M-4s/5s interactions result in narrower band-width and larger effective mass. With the increase of the atomic number in the same row of the periodic table, the M-4s/5s orbitals tend to be contracted. Assuming the M-M distance is the same, the M-4s/5s interactions should be weakened with the atomic number, in general. 

Figure 3. Schematic Illustration of dispersion curves of an acoustic phonon, a band electron, and the SWAP. The Incident phonon -q Is scattered as q2. (Reproduced with permission from reference 5. Copyright 1985 Nljhoff.)...

Lithium hydride, LiH, is a rather simple solid there are two atoms in the primitive cell and four electrons. It has the rock salt structure and its primitive cell contains one lithium and one hydrogen atom. Fig. 4.14. There are six dispersion curves of which three are acoustic and three optical. We can compare our calculations with the measured dispersion curves from LiD (rather than LiH because the deuterium coherent scattering cross section is required to measure the relative displacements of pairs of atoms ( 2.1). 

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