Conditions for formation of electron pairs in a metal

Abstract

In an isotropic model of the electron system of metal that is presented by the Fröhlich’s initial Hamiltonian, in the approximation of a weak electron–phonon interaction at T=0, first time, we show that the ground state of the system is the state with pairing correlations of electrons (the pair correlations of occupied electron states). In contrast to the BCS approach, the initial point in our approach is not electron pairing but is the maximum reduction of the energy of the considered system due to virtual processes of the electron–phonon interaction and to the exchange effect for the indirect electron–electron interaction (which is induced by certain phonon modes separately from others). In contrast to the BCS approach, we take into account the portion of the energy of the electron system that is connected with the above exchange effect. In the BCS approach, the corresponding portion is missed, and its role is prescribed to the portion that does not relate to the electron pairing.We show that expectation values of the above Hamiltonian for different wave functions for two interacting electrons above the Fermi sea of the non-interacting system (with interaction between the electrons that is induced by different phonon modes separately from others) are minimum for a certain structure of these functions and simultaneously for phonon modes that can induce the transitions of the interacting electrons between electron states in which they are (without violation of the Pauli exclusion principle and at everything else being equal). On the basis of the above results, the mechanism of the maximum reduction of the energy of the considered electron system is considered. In this mechanism each electron interact with the very different phonons, but in such way that give the maximum-possible negative contribution to the energy of the considered electron system.The theme of the article is conditions for the formation of electron pairs in a metal. This requires our understanding for the mechanism of the formation of electron pairs in a metal. The absence of this understanding is the main drawback of the BCS theory. The considered mechanism gives the solution. If this mechanism is feasible for a metal at T=0, the electron system of this metal can be described by the Hamiltonian that is similar to the BCS reduced Hamiltonian and the ground-state wave function is similar to the BCS ground-state wave function.The considered mechanism combines the simplicity and universality of the BCS model with giving wide opportunities to study conditions for the formation of the state of the electron system in a metal with the pair correlation of conduction electrons near the Fermi surface and with a gap in the spectrum of electronic excitations of this system and to study the dependence of these conditions on crystalline structure and structure of the conduction band of metals. It is so, because the considered mechanism has the universal nature but the above dependence is largely determined by the nature of virtual pairs in a metal.