The Electron-Phonon Interaction

The factor / indicates that F(r) is 90° out of phase with the local displacements. Such an electron potential, arising from phonons in crystals, is called an electron-phonon interaction. We saw that electrons may be freed in the crystal when impurities are present and may also be freed by thermal excitation even in the pure crystal. Any such free electrons contribute to the electrical conductivity, but that conductivity will in turn be limited by the scattering of the electrons by lattice vibrations or by defects. We will not go into the theories of such transport properties as electrical conductivity these arc discussed in most solid state physics texts- but will examine the origin of certain aspects of solids such as the electron-phonon interaction, which enter those theories. 

For the particular case of longitudinal optical modes, we found in Eq. (9-27) the electrostatic electron-phonon interaction, which turns out to be the dominant interaction with these modes in polar crystals. Interaction with transverse optical modes is much weaker. There is also an electrostatic interaction with acoustic modes -both longitudinal and transverse which may be calculated in terms of the polarization generated through the piezoelectric effect. (The piezoelectric electron phonon interaction was first treated by Meijer and Polder, 1953, and subsequently, it was treated more completely by Harrison, 1956). Clearly this interaction potential is proportional to the strain that is due to the vibration, and it also contains a factor of l/k obtained by using the Poisson equation to go from polarizations to potentials. The piezoelectric contribution to the coupling tends to be dominated by other contributions to the electron -phonon interaction in semiconductors at ordinary temperatures but, as we shall see, these other contribu- 

Similarly, the shift in the triply degenerate conduction-band levels are the negative of this and expressions for the shifts in any other band of interest are derivable from the formulae for the energy. 

Such electron-phonon interactions directly proportional to the dilatation are called deformation potentials, a concept first introduced by Bardeen and Shockley (see, for example, Shockley, 1950). This is indeed the dominant mechanism for electron-phonon interaction in covalent semiconductors, and the interaction with transverse waves is weaker. 

The qualitative validity of Eq. (9-29) can be checked by comparing the measured change in the gap between the energies of the valence and conduction bands at the center of the Brillouin Zone (k = 0) under pressure. (For a discussion, sec Paul and Waschauer, 1963). Paul and Waschauer (1963, p. 226) compiled experimental values of dEnl for Ge, GaAs, GaSb, InP, InAs, and InSb. They range from — 3 eV to —9 eV, without conspicuous trends, and where more than one measurement exists for one material, they differ by as much as a factor of two. The predictions based upon equations such as Eq. (9-29) vary from about — 2 eV to —4 eV. Thus the physical picture appears to be valid but it is not clear that it is quantitatively useful. 

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Figure 7. Phonon dispersion including the electron-phonon interaction for bcc CuZn. Force constants have been obtained from ah initio calculations. Dashed line is the phonon dispersion without the V-i contribution. Diamonds mark experimental data. ...

The generally accepted theory of electric superconductivity of metals is based upon an assumed interaction between the conduction electrons and phonons in the crystal.1-3 The resonating-valence-bond theory, which is a theoiy of the electronic structure of metals developed about 20 years ago,4-6 provides the basis for a detailed description of the electron-phonon interaction, in relation to the atomic numbers of elements and the composition of alloys, and leads, as described below, to the conclusion that there are two classes of superconductors, crest superconductors and trough superconductors. 

In this equation v is a phonon frequency, such that hv is approximately k, with the Debye characteristic temperature of the metal. The quantity p is the product of the density of electrons in energy at the Fermi surface, N(0), and the electron-phonon interaction energy, V. 

The gap in superconductivity between the fifth and sixth groups of the periodic table, discovered by Matthias,24 is seen to correspond to the transition from crest to trough superconductivity. It does not require for its explanation the assumption20- 25 that there are mechanisms of superconductivity other than the electron-phonon interaction. 

The theory of superconductivity based on the interaction of electrons and phonons was developed about thirty years ago. I 4 In this theory the electron-phonon interaction causes a clustering of electrons in momentum space such that the electrons move in phase with a phonon when the energy of this interaction is greater than the phonon energy hm. The theory is satisfactory in most respects. 

It is also shown that the electron-phonon interaction is operative in the polymerization process of the one-dimensional conjugated polymeric crystals a simple dynamical model for the polymerisation in polydiacetylenes is presented that accounts for the existing observations. 

As was mentioned above, the motion of one electron in these systems affects the motion of other electrons in the system. Therefore, the electron - phonon interaction can be said to mediate electron... 

The next important aspect to be considered is the electron-phonon interaction (lattice relaxation). Here, the effect of momentum conserving phonons, or promoting modes, can in principle be included in the electronic cross section this is discussed, for instance, by Monemar and Samuelson (1976) and Stoneham (1977). However, the configuration coordinate (CC) phonons (or accepting modes) are treated separately. The effect of these CC modes is usually expressed by the Franck-Condon factor dF c, where this factor is the same as the defined in our Fig. 16. Thus assuming a single mode,... 

The 0 e Renner-Teller vibronic system describes an orbital doublet ( ) interacting with a two-dimensional vibrational mode of s symmetry. In Section 2 we determine the general formal structure of the electron-phonon interaction matrices with orbital electronic functions of different symmetry (p-like, d-like, /-like, etc.), exploiting their intuitive relation with the Slater-Koster matrices of the two-center integrals. A direct connection with the form obtained through the molecule symmetry is discussed in Section 3. 

In considering the vibronic side-bands to be expected in the optical spectra when we augment the static crystal field model by including the electron-phonon interaction, we must know the frequencies and symmetries of the lattice phonons at various critical points in the phonon density of states. We shall be particularly interested in those critical points which occur at the symmetry points T, A and at the A line in the Brillouin zone. Using the method of factor group for crystals we have ... 

The key to understand the anomalous behaviors in A vC,o is the electron-electron interaction, U, and the electron-phonon interaction, S. In A vCgo, the important interaction as U is the Coulomb repulsion between the flu electrons. The importance of U has been pointed out based on the results of the photoemission experiments [15]. Also, in AxC o, the important interaction as S is the coupling of the tlu electrons to the intramolecular phonons of the CRaman experiments [16], Nevertheless, a complete understanding of the anomalous behaviors in AxCgo has not been established yet. The reason is that the system with the orbital degree of freedom in which both U and S are important has not been known so far [17-20],... 

The purpose of the present study is to show a unified picture for understanding the electronic properties of A Qo based on a simple model. In this model, both the electron-electron interaction U and the electron-phonon interaction S are taken... 

In this section, a model which gives the basis of the present study is introduced to investigate the electronic properties of A,Cfio [17]. First, the one-electron part of the Hamiltonian, which describes the itinerant motion of the flu electrons in terms of the electron transfer T, is given. Next, the electron-electron interaction U and the electron-phonon interaction S are examined U represents the Coulomb repulsion between the t u electrons and S represents the coupling of the fiu electrons to the intramolecular phonons of the Cdynamical aspect of S is pointed out. 

Taking removes the degeneration of the vibronic states breaking the rotational symmetry of the electron-phonon interactions, the model still staying within the class of JT models [1,2]. 

The electronic Hamiltonian He is now augmented by the electron-phonon interaction [61],... 

Given the lattice Hamiltonian Eq. (5), which casts the interactions in terms of site-specific and site-site interaction terms, a complementary diabatic representation can be constructed which diagonalizes the Hamiltonian excluding the electron-phonon interaction, Hq = He + f7ph. This leads to the form... 

Following the analysis of Refs. [54,55,72], we now make use of the fact that the nuclear modes of the Hamiltonian Eq. (8) produce cumulative effects by their coupling to the electronic subsystem. From Eq. (9), the electron-phonon interaction can be absorbed into the following collective modes,... 

Related diborides (ZrB2, NbB2, and TaB2) have PC spectra proportional to the electron-phonon-interaction spectral function, like... 

The electron-phonon interaction has been studied also in a LiTnJA crystal by Kupchikov et al. (1982). They have measured Raman and infrared reflection spectra under pressures up to 1.2 GPa and at temperatures ranging from 4.2 K to 300 K. The interaction of optical phonons with electronic excitations in this system of rare-earth ions was detected by anomalous tem-... 

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