Electron-phonon matrix elements

The zero-approximation in expansion of I(V) in d/l is the ohmic current considered by Sharvin [5]. From the Sharvin s formula the characteristic size d of the contact can be determined in the ballistic limit. The second derivative of the first approximation in expansion of I(V) in d/l is directly proportional to the spectral function of electron-phonon interaction (PC EPI) gpc w) = apc (w) F (w) °f the specific point-contact transport both in the normal and in the superconducting states [1, 6, 7], This term is the basis of the canonical inelastic point-contact spectroscopy (PCS). Here, ot2pC (oj) is the average electron-phonon matrix element taking into account the kinematic restriction imposed by contact geometry and F (oj) is the phonon density of states. 

The fact that the bare coupling is short-ranged, or more precisely, that its anisotropy is roughly the same as that of the electronic band, is important in order to understand the very existence of quasi-one-dimensional materials. If the electron band is quasi-one-dimensional, such will also be the bare electron-phonon matrix element. E. g. band... 

Na Is the mass of atom a and la the square of an electron-phonon matrix element. Expressions for T(. have been given by McMillan (31 and by Allen and Dynes (12) which are functions of X and ii, a... 

The effective mass ratios measured are of the order of one. The deformation potential coupUng constants vary between 0.5 x 10 K and 3.8 x 10 K. That deduced from the temperature dependence is 10 K. From the band structure for LaAg it was conjectured that the phase transition in the LaAgIn compounds could be due to a nesting feature of the Fermi surface, which gives large electron-phonon matrix elements for the observed M-point phonons (Knorr et al. 1980, Niksch et al. 1987). 

Herbert et al. have used a similar model to calculate electron-phonon matrix elements (but not hole-phonon) for a number of transitions in Ge, Si, GaAs and InP. These values have been compared to experiment, particularly transport measurements. In general there is good agreement. However, such comparisons are not as sensitive as those made for the indirect gap measurements, both stressed and unstressed, since in this case there is a delicate balance in the interference between EP and HP terms. 

Although Herbert et al. have calculated a number of intervalley electron-phonon matrix elements there is overlap with our theoretical determinations in the case of T-L of Ge (LA-phonon). However, from Herbert s work it is not clear whether he has considered both intermediate states (i.e., I2 c 5 g). If we assume that he has taken into account only 12 q then there is fairly good agreement between his values and ours (see Table IX). 

The hole burning data obtained for P870 establish that the P870 lifetime (from total zero-point) is invariant to excitation frequency within the heterogeneity-induced zero-phonon line frequency distribution of P870. This is consistent with, but does not prove, an absence of dispersive kinetics from a distribution of values for the pure electronic coupling matrix element V. The proposed studies mentioned above may allow for a definite conclusion to be reached on this point. 

The ab initio calculation of the transition rate between two electronic states with the emission of p phonons involves a very complicated sum over phonon modes and intermediate states. Due to this complexity, these sums are extremely difficult to compute however, it is just this complexity, which permits a very simple phenomenological theory to be used. There are an extremely large number of ways in which p phonons can be emitted and the sums over phonon modes and intermediate states are essentially a statistical average of matrix elements. In the phenomenological approach it is assumed that the ratio of thep-th and (p - l)-th processes will be given by a couphng constant characteristic of the matrix in which the rare earth is situated and not depending... 

Consider now the equality Hoj> n Jm=8j>j. Thus, in this model, preparing the system in the ground state of the Coulomb Hamiltonian, no time evolution can be expected if we do not switch on the kinematic couplings. We take a simple case where the electron-phonon coupling is on. The matrix elements of H in this base set look like ... 

The term E%(eq) is the potential energy of the clumped nuclei configuration at the equilibrium geometry R0 (crude level), hpQ is one -electron term ( core Hamiltonian ) for equilibrium nuclear configuration Ro, and upq (Q) represents matrix element of electron - vibration (phonon) coupling. 

The electron-phonon coupling term is constructed so as to account for the modulation of the band gap - stemming from the diagonal single-particle matrix elements of Eq. (4) - as a function of the (la) phonon modes belonging to the oIh phonon branch and Zth site [29],... 

This Hamiltonian is similar to the usual electron-phonon Hamiltonian, but the vibrations are like localized phonons and q is an index labeling them, not the wave-vector. We include both diagonal coupling, which describes a change of the electrostatic energy with the distance between atoms, and the off-diagonal coupling, which describes the dependence of the matrix elements tap over the distance between atoms. 

Now, the non-adiabatic electron transitions is examined only when electron matrix element Fif is small (see the criterion (10) and (10a)). It is the criterion of applicability of the perturbation theory on F f, but it is not the criterion of applicability of the concept of non-adiabatic transition between two crossing diabatic terms. As it is known (see, for example, ref. [5]) the true image of terms is changed on taking into account the interaction V. Denote two terms without inter-term interaction as E[(R) and E (R), where R is the generalized nuclear coordinate. If the crystal phonons (or the outer-sphere variables in a polar medium) only participate in the transition, then E[(R) and E (R) are the parabolic terms independent of the value of shift of... 

Some important problems of the theory of multi-phonon electron transition were not touched upon in this chapter. These are, first, the calculation of the expression for the electron matrix element at the tunneling transfer, second, the influence of medium on the electron matrix element, and, finally, the investigation of the applicability of Born-Oppenheimer approach in the electron tunneling transfer. These issues will be considered in the next chapter. 

Wavefunctions and Charge Distributions. Though the quality of the wavefunction obtained in a crystal orbital study cannot be assessed by direct comparison with experiment it is of decisive importance from the point of view of prospective transport calculations on conducting polymers (calculation of electron-phonon interaction matrix elements, optical properties, etc.). Of course, the wavefunction also plays a fundamental role when properties related to the many-electron energy are calculated, and therefore the quality of these quantities partially characterizes that of the wavefunction. 

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