Momentum crystal

Eig. 1. Representation of the band stmcture of GaAs, a prototypical direct band gap semiconductor. Electron energy, E, is usually measured in electron volts relative to the valence, band maximum which is used as the 2ero reference. Crystal momentum, is in the first BriUouin 2one in units of 27r/a... 

PL is generally most usefril in semiconductors if their band gap is direct, i.e., if the extrema of the conduction and valence bands have the same crystal momentum, and optical transitions are momentum-allowed. Especially at low temperatures. 

Graphite exhibits strong second-order Raman-active features. These features are expected and observed in carbon tubules, as well. Momentum and energy conservation, and the phonon density of states determine, to a large extent, the second-order spectra. By conservation of energy hut = huty + hbi2, where bi and ill) (/ = 1,2) are, respectively, the frequencies of the incoming photon and those of the simultaneously excited normal modes. There is also a crystal momentum selection rule hV. = -I- q, where k and q/... 

An important and interesting question is obviously whether for quasicrystals and incommensurately modulated crystals there is anything corresponding to the Bloch functions for crystals. Momentum space may be a better hunting ground in that connection than ordinary space, where we have no lattice. Not only is there no lattice, one cannot even specify the location of each atom yet [8]. 

Thus, in addition to the dispersion itself, we get information about which band is occupied in which Brillouin zone. This is a consequence of the fact that EMS measures real momentum, and not, like for example angle-resolved photoemission, crystal momentum. 

Bulk silicon is a semiconductor with an indirect band structure, as schematically shown in Fig. 7.12 c. The top of the VB is located at the center of the Brillouin zone, while the CB has six minima at the equivalent (100) directions. The only allowed optical transition is a vertical transition of a photon with a subsequent electron-phonon scattering process which is needed to conserve the crystal momentum, as indicated by arrows in Fig. 7.12 c. The relevant phonon modes include transverse optical phonons (TO 56 meV), longitudinal optical phonons (LO 53.5 meV) and transverse acoustic phonons (TA 18.7 meV). At very low temperature a splitting (2.5 meV) of the main free exciton line in TO and LO replicas can be observed [Kol5]. 

Multiplying the wavevector k by Planck s constant h and dividing by 2n yields the crystal momentum p = hk. This crystal momentum of the electron includes a lattice component and is therefore not a true free-particle momentum, as it was in free-electron theory. 

If dE/dk=/= 0, then Umklapp scattering will occur The crystal momentum phonon will and must "borrow" momentum and energy from the reciprocal lattice ... 

It is important to note that in solids distances between nearest atoms can vary in different directions, and hence the minimum of the valley may not occur at kx=ky = k = 0, but at some point defining a specific direction, as shown in Figure 2.4B for a crystalline solid. In an optical transition, both energy and momentum must be conserved. Because the momentum of a photon, h/X (X is the wavelength of light which is typically thousands of angstroms), is very small compared to the crystal momentum h/a (a is the lattice constant, typically a few Angstroms), the photon-absorption process should conserve the electron momentum. 

Because the photon momentum is negligible compared to the electron crystal momentum, the momentum conservation requirement simplifies to ... 

Since the momentum of photons, h/A, is small compared with the crystal momentum, hla (a is the lattice constant), the momentum of electrons should be conserved during the absorption of photons. The absorption coefficient a hv) for a given photon energy is proportional to the probability, P, for transition from the initial to the final state and to the density of electrons in the initial state as well as to the density of empty final states. On this basis, a relation between absorption coefficient a and photon energy ph can be derived [2, 4]. For a direct band-band transition, for which the momentum remains constant (see Fig. 1.7), it has been obtained for a parabolic energy structure (near the absorption edge) ... 

If N -> oo the matrices , , X, Y and ft are of infinite dimension. Taking into account the periodic symmetry of the polymer the infinite sums over (crystal momentum) can be transformed into an integration over in the first Brillouin zone giving... 

The eigenvectors of the lattice periodic Hamiltonian of the Kohn-Sham-Dirac equation (28) are Bloch states fen) with crystal momentum k and band index n. They are expressed by the ansatz... 

In the quantum mechanical treatment of this model, the equations of motion in the harmonic approximation become analogous to those for electromagnetic waves in space [2-4]. Thus, each wave is associated with a quantum of vibrational energy hu and a crystal momentum hq. By analogy to the photon for the electromagnetic quantum, the lattice vibrational quantum is called a phonon. The amplitude of the wave reflects the phonon population in the vibrational mode (i.e., the mode with frequency co and... 

An analogous expression is obtained in three dimensions. We now need to consider periodic systems. As we have discussed, the wavefunction for a particle on a periodic lattice must satisfy Bloch s theorem. Equation (3.85). The wavevector k in Bloch s theorem plays the same role in the study of periodic systems as the vector k does for a free particle. One important difference is that whereas the wavevector is directly related to the momentum for a free particle (i.e. k = p/h) this is not the case for the Bloch particle due to the presence of the external potential (i.e. the nuclei). However, it is very convenient to consider Hk as analogous to the momentum and it is often referred to as the crystal momentum for this reason. The possible values that k can adopt are given by ... 

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