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Superconductor Science and Technology

Pairing instability in an interacting Fermi gas

Y. M. Malozovsky, J. D. Fan

Published May 1, 1997• 14 citations
Publisher

Abstract

The Cooper instability in an interacting Fermi gas is examined by using the perturbational diagram approach. A graphical derivative method based on Ward's identity is introduced to represent the contributions to the interparticle interactions from different channels. It is shown that pairing instability for the electrons excited from the Fermi sea may happen as a result of the presence of holes even when the original interaction is repulsive. Thus, in contrast to the Cooper instability caused by a given attractive interaction, the pairing instability can be induced by the particle - hole excitations. The application of the results to a layered two-dimensional Fermi gas is considered.

DOI
10.1088/0953-2048/10/5/002

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The Cooper instability in an interacting Fermi gas is examined by using the perturbational diagram approach. A graphical derivative method based on Ward's identity is introduced to represent the contributions to the interparticle interactions from different channels. It is shown that pairing instability for the electrons excited from the Fermi sea may happen as a result of the presence of holes even when the original interaction is repulsive. Thus, in contrast to the Cooper instability caused by a given attractive interaction, the pairing instability can be induced by the particle - hole excitations. The application of the results to a layered two-dimensional Fermi gas is considered.
The pairing instability induced by either a given attractive interaction or particle-hole excitations is examined by using the perturbational diagram approach. A graphical functional derivative method based on Ward's identity is developed to represent the contributions to the interparticle interactions from different channels. It is shown that pairing instability for the electrons excited from the Fermi sea may happen due to the presence of holes even when the original interaction is repulsive. Thus, in contrast to the Cooper instability caused by a given attractive interaction, the pairing instability can be induced by the particle-hole excitations. The scattering diagrams from this model allows one to evaluate the two particle vertex function and Tc by using the pole condition of the vertex function. It has been found that Tc can reach quite high values and gives rise to the bell shape. The application of the results to a layered 2D Fermi gas is considered.
The Cooper instability in a Fermi gas is examined using the perturbative diagram approach. A graphical functional derivative technique based on Ward's identity is developed to obtain two-particle interactions and then to calculate the vertex part. The pairing instability for a given interaction, such as a phonon (plasmon, etc.) field, occurs in a quiescent Fermi sea, i.e. without exciting or involving background particles (holes), only if the interaction is attractive, as first proposed by Cooper and adopted in the BCS (Bardeen–Cooper–Schrieffer) theory. The consequence from this technique provides a way to evaluate the effect of the vertex corrections and responses of the Fermi gas in both charge and spin channels incorporating the backward scattering process. The significance of the methodology presented in the present work lies in the fact that it can both reproduce the known results, and, more importantly, be extended to investigate the intermediate or strong coupling case, such as nuclear interactions, where a neglect of vertex corrections may not be a good approximation.
By means of the Fermi-liquid approach, the effects of exchange and correlations are considered in a layered two-dimensional electron gas. A graphical derivative technique is introduced to obtain the contributions to the interparticle interactions from different channels. It is shown that the interaction in the particle - hole channel can induce the metal - insulator phase transition when the Fermi momentum is comparable with the inverse interlayer spacing, . It is also shown that the Cooper-like scattering processes give rise to the scaling of the Fermi liquid response as Im , which leads to the linear temperature dependence of the quasiparticle damping. The possibility of superconductivity in the region between the insulator and normal metal phases, accounting for the particle - hole excitations, is discussed. The effect of the spin fluctuations on the superconductivity due to the particle - hole excitations is also considered.
We consider possible superconducting instabilities in a two-dimensional Fermi system with short-ranged repulsive interactions between electrons. The possibility of an unusual superconducting paring due to the Kohn-Luttinger mechanism is examined. The quasiparticle scattering amplitude is shown to possess an attractive harmonic in second-order perturbation theory for finite values of the energy transfer. The corresponding singularity in the pairing vertex leads to a superconducting pairing of the electron excitations with finite energies. We identify the energy transfer in the Cooper channel as the binding energy of the excited pair. At low enough temperatures, the Fermi system is a mixture of normal electron excitations and fluctuating d-wave Cooper pairs possessing a finite gap.
Relation between the binding energy of a pair of electrons in Cooper's configuration and the magnitude of the attractive interaction between them is investigated theoretically for a two-dimensional electron gas when the electron pair has (a) zero and (b) finite center of mass momentum (CMM) first (i) in the background of a passive Fermi sea and (ii) with inclusion of the effect of Pauli exclusion principle in the presence of an active Fermi sea, at zero temperature. In the Cooper's approach (Cooper 1956 Phys. Rev. 104 1189) two pairing electrons are considered above the Fermi sea. Due to Pauli exclusion principle, these two electrons are forbidden to go into the states whose energies are less than the Fermi energy. Consequently, the role of the electrons which form the Fermi sea is passive in formation of electron pair (Fujita and Godoy 2001 Theory of High Temperature Superconductivity (Kluwer Academic Publishers)). However, in the case of an active Fermi sea, implying possible presence of multiple pairs of electrons coming out of the Fermi sea, the two electrons will have less available region in the momentum space for pair formation due to the Pauli blocking effect. In both the situations, threshold values of attractive interaction are found to exist to bind the electrons having finite CMM even with respect to the Fermi energy while with arbitrarily small value of attractive interaction they can be bound only in the case of zero CMM and that too in the presence of a passive Fermi sea. However, the presence of an active Fermi sea strongly opposes the pair formation having lower CMM. Furthermore, raising the range of the attractive interaction below and above the Fermi surface makes the pair formation more difficult due to enhanced Pauli blocking in this case. In the case of a passive Fermi sea however, with the increase in the range of the attractive interaction it becomes easier for the electron pairs to be in both Cooper pair state (CP) and true bound state (TB). These results are totally distinct from those obtained by Randeria et al (1989 Phys. Rev. Lett. 62 981, 1990 Phys. Rev. B 41 327) that a fermionic bound state in true vacuum is a necessary and sufficient condition for the Cooper pairing instability in the presence of a passive Fermi sea, independent of attractive interaction strength implying that for a potential which is attractive everywhere in two dimension, a two-body bound state exists for an arbitrarily weak attraction. Our calculations show, in the mean field approximation, that co-existence of TB state and CP state with various CMMs can occur only beyond a critical attractive interaction strength. This could lead to the formation of incoherent pairs even under strong coupling situation, which may hint onto the pseudo gap formation leading to some of the exotic features in the anomalous normal state seen in some of the exotic superconductors.
Weak attractive interactions in a spin-imbalanced Fermi gas induce a multiparticle instability, binding multiple fermions together. The maximum binding energy per particle is achieved when the ratio of the number of up- and down-spin particles in the instability is equal to the ratio of the up- and down-spin densities of states in momentum at the Fermi surfaces, to utilize the variational freedom of all available momentum states. We derive this result using an analytical approach, and verify it using exact diagonalization. The multiparticle instability extends the Cooper pairing instability of balanced Fermi gases to the imbalanced case, and could form the basis of a many-body state, analogously to the construction of the Bardeen-Cooper-Schrieffer theory of superconductivity out of Cooper pairs.