On the Electron-Phonon Interaction in Normal Metals. I

Abstract

By the use of Frohlich's Hamil!onian and the method of Green's functions, the electron­ phonon interaction in a normal metal under a steady magnetic field is studied. It is shown that the period of the de Haas-van Alphen effect is not modified by the interaction. The resonance frequency of the Azbel'-Kaner effect, on the other hand, is modified and the cyclotron mass of the electron takes essentially the same value as the thermal mass which is also modified by the interaction as shown previously. The conclusion is in accord with the recent experimental results obtained by Kip and Grimes on Na and K. The same conclusion, however, is obtained also by applying Landau's theory of Fermi liquids to the inter-electronic Coulomb correlation without introducing the electron-phonon interaction, though the mass shift caused by the former seems too small. § I. Introduction In a previous paper'> we examined how the electron-phonon interaction modifies the density of one-electron states (i.e. the thermal mass of the electron) and electric conductivity of a normal metal. As emphasized there, the physical point is the dynamical character of the interaction. This character is already evident in the lowest order process in the perturbation theory of the electron self-energy. Thus, between emission and reabsorption of a phonon by an elec-. tron, there exists retardation which makes the time dependence of the self­ energy essential. Indeed, as we have shown, this dynamical part of the self­ energy becomes singular when the velocity of sound v. is much less than the electron velocity Vp at the Fermi surface and appreciably modifies the thermal mass. It results also in some anomalous dispersion of the one~electron excita­ tion spectrum when excitation energies are comparable to the Debye temperature. On the other hand, in the case of electric conductivity, the renormalization of the wave function explicitly comes into play in addition to the mass renormali­ zation mentioned above. These two effects are cancelled by each other both in static and anomalous limits and there appears the bare (band theoretical) electron mass in the expression for conductivity. The aim of the present paper is to extend the theory so as to include a steady magnetic field. We thus deal with de Haas-van Alphen effect and Azbel'­ Kaner effect, taking into consideration the electron-phonon interaction. As for the de Haas-van Alphen effect, Luttinger 2> and Gor'kov and Bychkov3> have already