Dispersion in k-space

Semicore states, with relatively low energies, which show only a slight energy dispersion in k space. They are usually not displayed in the band structure. 

The dynamic problem of vibrational spectroscopy must be solved to find the normal coordinates as linear combinations of the basis Bloch functions, together with the amplitudes and frequencies of these normal vibrations. These depend on k, and therefore the problem must be solved for a number of k-points to ensure an adequate sampling of the Brillouin zone. Vibrational frequencies spread in k-space, just as the Bloch treatment of electronic energy gave a dispersion of electronic energies in k-space. The number of vibrational levels whose energy lies between E and fc +d E is called the vibrational density of states. Vibrational contributions to the heat capacity and to the crystal entropy can be calculated by appropriate integrations over the vibrational density of states, just like molecular heat capacities and entropies are obtained by summation over molecular vibration frequencies. 

We would like now to focus on initial conditions for the electron ensemble immediately after the optical excitation. The laser pulse may excite electron-hole pairs in the middle of the Wannier-Stark ladder, lower or above which within the miniband picture corresponds to excitation of electrons in the middle, bottom or top of the miniband, correspondingly. We assumed that all electrons were initially distributed in k-space in accordance with the normal distribution /(A J=(l/V2 o-i ) exp(-( -( j)2/2<7t2) where (k is the average k, and [Pg.201]

It is also possible to obtain whole phonon dispersion curves along some direction in k-space from the ab initio calcula-tions." - This can be done in two equivalent ways. One way is... 

Core states, which lie energetically deep and are not directly involved in the bonding. They can be treated as atomic-like and calculated using the self-consistent crystal potential. They show no dispersion of the energy in k space. 

Actually, there is not only a nesting condition in k space, but also in the dispersion relation, see Giamarchi, T. 

Fig.4. Schematic representation of the different common phase-matching techniques in the k space representation. (ADM) anomalous dispersion (WBM) waveguide birefringence (MD) modal dispersion (QPM) quasi-phase-matching (C) Cerenkov and (CP) counter propagating Cerenkov...

The dispersion relation co(k) is the relation between the angular frequency co (time-domain behavior) and the wavevector k (space-domain behavior). Dispersion means that waves of different angular frequencies co can travel at different speeds v. In the present simple case the dispersion relation is the linear relationship ... 

The dispersion relationships of lattice waves may be simply described within the first Brillouin zone of the crystal. When all unit cells are in phase, the wavelength of the lattice vibration tends to infinity and k approaches zero. Such zero-phonon modes are present at the center of the Brillouin zone. The variation in phonon frequency as reciprocal k) space is traversed is what is meant by dispersion, and each set of vibrational modes related by dispersion is a branch. For each unit cell, three modes correspond to translation of all the atoms in the same direction. A lattice wave resulting from such displacements is similar to propagation of a sound wave hence these are acoustic branches (Fig. 2.28). The remaining 3N-3 branches involve relative displacements of atoms within each cell and are known as optical branches, since only vibrations of this type may interact with light. 

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