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Journal of Physics Condensed Matter

The Cooper pair problem for generalized Fermi surfaces

MANUEL DE LLANO, James P. Vary

Published Jan 21, 1991• 2 citations
Publisher

Abstract

The Frohlich second-order perturbative treatment of the electron-phonon system with a generalized Fermi sea is extended to infinite order by solving the Cooper electron-pair problem in that sea. Substantially tighter-bound pairs follow for fixed coupling no matter how weak.
DOI
10.1088/0953-8984/3/3/007

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The Cooper pair eigenvalue equation is solved exactly for the free-electron density of states in one, two, and three dimensions. The two-dimensional (2-D) case is essentially the original derivation of Cooper, as is the three-dimensional case if the interaction is restricted to the vicinity of the Fermi surface. The essential singularity in coupling, familiar in 2-D quantum binding in a well, and sometimes erroneously attributed to the two-dimensional character of the BCS model interaction which is nonzero only very near the Fermi surface, appears in all three cases.
A recently [9] proposed method of continuous unitary transformations is used to eliminate the interaction between electrons and phonons. The differential equations for the couplings represent an infinitesimal formulation of a sequence of Fröhlichtransformations [5]. The two approaches are compared. Our result will turn out to be less singular than Fröhlich’s. Furthermore the interaction between electrons belonging to a Cooper-pair will always be attractive in our approach. Even in the case where Fröhlich’s transformation is not defined (Fröhlich actually excluded these regions from the transformation), we obtain an elimination of the electron-phonon interaction. This is due to a sufficiently slow change of the phonon energies as a function of the flow parameter. 1 New address: Max-Planck-Institut für Kolloid- und Grenzflächenforschung, Kantstraße 55,