Electron wave vector

In direct gap GaAs, an excited electron at the bottom of the conduction band can relax spontaneously back into a hole in the valence band by emitting a photon at the band gap energy. This electron-hole radiative recombination process can only occur in Si if momentum is conserved, i.e., the excited electron wave vector must be reduced to zero. This, in pure Si, occurs via the transfer of momentum to a phonon that is created with equal and opposite wave vector to that of the initial state in the conduction band. Such a three-body process is quite inefficient compared with direct gap recombination.1218 This is why Si is such a poor light emitter. 

In practice there are three main difficulties in the use of this technique. The 2kF and 4kF scattering is not observed in all materials or at all temperatures. Fortunately, pretransitional effects (one-dimensional fluctuations) are often observed in a wide temperature range above Tc [114]. Second, the corresponding scattering is often very weak and requires special measurement techniques. For example, in the case of TTF-TCNQ, the intensity of 2kF satellites is 10 4 that of main reflections and the onedimensional scattering at 60 K is a further 10 2 to 10 3 weaker [133]. Third, to get p, one must decide whether the observed instability occurs at the 2kF or at the 4kF electronic wave vector. 

Equation 7.5 describes the energy-dependent EXAFS oscillations that can be interpreted using the EXAFS equation (as given in Equation 7.1 or Equation 7.2). X(k) EXAFS as a function of electron wave vector. 

The variation with the electron wave vector k is associated with an intensity variation in the experimentally observed polar and azimuth angles. (In order to include vibrational attenuation of interference effects, each scattered wave has to be multiplied by the temperature dependant Debye-Waller factor.)... 

For elastic scattering, the absolute value of the scattered and the incident electron wave vector are equal, given by the reciprocal value of the wavelength, k = ko = 1/A. The angle between scattered and incident wave vector is the scattering angle 9 and the vector q, given by... 

Fig. 3.4. Band structure of germanium calculated from empirical pseudopotentials including s-o coupling as a function of the electron wave vector along selected directions of the reciprocal lattice. The s-o splitting Aso is the energy difference between Ts+ and IV. The light- and heavy-hole VBs are the IV1" to Le and IV1" to L4 +L5 bands, respectively. The energy reference is the VB maximum at the T point after [21]...

An electron hole pair is created in a semiconductor when a photon of sufficient energy is absorbed, resulting in excitation of an electron from the valence band to the conduction band [115]. In the context of semiconductor photoelectrochemistry, it is useful to distinguish between direct and indirect optical transitions. If the top of the valence band and the bottom of the conduction band are both situated at = 0 (A being the electron wave vector), one-step optical processes between delocalised states in the valence and conduction band can occur. The absorption coefficient for direct absorption of photons of energy hv, in a semiconductor with bandgap Eg is given by... 

The optical band gap of the semiconductor (Sect. 2) is an important parameter in defining its light absorption behavior. In this quantized process, an electron-hole pair is generated in the semiconductor when a photon of energy hv (v = frequency and hv > Eg) is absorbed. Optical excitation thus results in a delocalized electron in the CB, leaving behind a delocalized hole in the VB this is the band-to-band transition. Such transitions are of two types direct and indirect. In the former, momentum is conserved and the top of VB and the bottom of CB are both located at k = 0 (k is the electron wave vector). The absorption coefficient (a) for such transitions is given by [106]... 

Figure 2.8 (a) A three-dimensional crystal in the real-space Lx, Ly, Lz are dimensions of the crystal, (b) The electron wave vector k in fe-space. The components of fe are kx = TtnxjLx] ky — JUMy/Lyl kz ---... 

Only phonons with q less than 2k k is the absolute value of the electron wave vector) can interact with electrons. In nondegenerate semiconductors the wave vector of most electrons is much smaller than the wave vector of the thermal phonons at not very low temperatures. Therefore, only the longest-wavelength phonons can interact with electrons giving rise to a small contribution to the thermal conductivity. 

The electron wave vectors at the Fermi level in metals are more likely, rather than the wave vectors of phonons, to make a significant contribution to the heat transport. These phonons can interact with electrons, and thus, an effect of phonon-electron scattering is to be expected in metals at all temperatures. One fails, however, to observe phonon-electron scattering in pure metals at high temperatures because of the complexity of the Kl separation. One can separate and and observe phonon-electron scattering in rare earth metals and metal-like compounds which contain rare earth elements, due to the low mobility of electrons which increases p considerably and hence decreases k ) (Oskotski and Smirnov 1971, Khusnutdinova et al. 1971, Luguev et al. 1975a). 

Assuming translational symmetry parallel to the surface of crystalline or polycrystalline material, the electron wave vector of the final state perpendicular to the surface, k/, is smaller due to the difference of the final state energy, Ef, and the work function, O, of the sample. The emission cone is given by the angular condition,... 

Schematic drawing of scattering at a single center (atomic nucleus), showing the change of momentum s as a difference vector between the incoming and outgoing electron wave vectors k and fc. ...

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