## 5 Intervalley Scattering and Franck-Condon Transition
### On- and Off-Resonance Effects  
In this section, let us discuss the on- and off-resonance effects on the present intervalley scattering process and its experimental observation. The observation of the L valley electron can be seen as a two-step process naively. In the first process, the electrons in the ground state are excited into the \(\Gamma\) valley by a visible light and scattered to the final L valley after a while. In the second process, the L valley electron is emitted out of the crystal by the second light excitation.  
In the first process through the elastic e-ph scattering, the matrix element between the intermediate \(\Gamma\) valley state and the final L valley one is given by the spatial randomness of the ground state phonons. As discussed in the preceding section, since this transfer of electrons from the \(\Gamma\) valley to the L one is a diffusion within an eigenstate, the elastic intervalley scattering has the same order as the Rayleigh scattering as an optical process. So far, we have treated only the on-resonant process in the elastic e-ph scattering, in which the photo-excited electrons are scattered to the L valley without a change of the energy, as shown in Figure 8b. As a general theory, the off-resonant process is also possible, while the intensity by the off-resonant process is expected to be much weaker than that of the resonant process in many cases. However, on- and off-resonance effects are basically indistinguishable as a final state interaction of light absorption.  
In the second process, the initial L valley electron is emitted out of crystal by the second visible light excitation. In this process, the L valley electron can pass through the unoccupied states in the  
Figure 12: Time constant by the elastic \(\Gamma\)-L e-ph transition, including the non-Condon effect, is given.  
conduction band by the elastic e-ph process, as shown in Figure 13. This fact is reflected in the transition rate as the initial state interaction. Then, the initial state of this process would be written as:  
\[\mid\mathbf{k}\rangle+\frac{1}{N}\sum_{\mathbf{q}}\frac{V_{\mathbf{k},\mathbf{k}-\mathbf{q}}}{E(\bm {k})-E(\mathbf{k}-\mathbf{q})}\mid\mathbf{k}-\mathbf{q}\rangle, \tag{29}\]  
where \(\mid\mathbf{k}\rangle\) is the electronic state with a wave vector \(\mathbf{k}\) and \(V_{\mathbf{k},\mathbf{k}-\mathbf{q}}\) is the coupling constant of the elastic e-ph interaction between \(\mathbf{k}\) and \(\mathbf{k}-\mathbf{q}\). In an actual experimental observation [10], the observed kinetic energy of the photo-emitted electron from the L valley is the same regardless of whether the electron passed through the above intermediate state or not. On the other hand, the momentum of the electron reflects their path. As a result, intensities along the equi-energy line, in which there is no one-body state, may be able to appear. In fact, we can see intensities between the \(\Gamma\) valley and the L one along the equi-energy line in the experimental EDF (Figure 14).  
Figure 14: The experimental energy distribution function (EDF) in the conduction band of GaAs, obtained by two-photon photoemission spectroscopy (2 PPES). The solid and dashed curves show the band dispersion along \(\Gamma\)-L and \(\Gamma\)-X directions, plotted as a function of angle, based on the band-structure calculation in Ref. [10]. Adapted with permission form [10], copyright American Physical Society, 2016. For more details, see [10].  
Figure 13: A schematic view of the initial state interaction in the photo-emission process of the L valley electrons. The colored arrows show some possible emission paths of a L valley electron, passing through the unoccupied states in the conduction band. For more details, see the text.  
It should be noted that intermediate state can also be realized through other mechanisms such as elastic e-e scattering. However, the intensity would be much weaker than that by the elastic e-ph scattering.