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Superconductor Science and Technology

The Cooper problem in nanoscale: enhancement of the coupling due to confinement

Mihail D Croitoru, Alexei Vagov, Arkady A Shanenko, V. M. Axt

Published Nov 8, 2012• 10 citations
Publisher

Abstract

In 1956 Cooper demonstrated (1956 Phys. Rev. 104 1189) that, no matter how weak the attraction is, two electrons in three-dimensional (3D) space just above the Fermi sea could be bound. In this work we investigate the influence of confinement on the binding energy of a Cooper pair. We show that confinement-induced modification of the Fermi sea results in a significant increase of the binding energy, when the bottom of an energy subband is very close to the Fermi surface.

DOI
10.1088/0953-2048/25/12/124001

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arXiv:2408.17015•SciPost Physics•Dec 11, 2024•1 citation
In 1956 Cooper demonstrated (1956 Phys. Rev. 104 1189) that, no matter how weak the attraction is, two electrons in three-dimensional (3D) space just above the Fermi sea could be bound. In this work we investigate the influence of confinement on the binding energy of a Cooper pair. We show that confinement-induced modification of the Fermi sea results in a significant increase of the binding energy, when the bottom of an energy subband is very close to the Fermi surface.
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.
This work deals with the key constituent behind the existence of superfluid states in ultracold fermionic gases confined in a harmonic trap in 2D, namely, the formation of Cooper pairs in the presence of a Fermi sea in inhomogeneous confinement. For a set of finite-range models representing particle–particle interaction, we first ascertain the simultaneity of the emergence of bound states and the divergence of the s-wave scattering length in 2D as a function of the interaction potential parameters in free space. Then, through the analysis of two particles interacting in 2D harmonic confinement, we evaluate the energy shift with respect to the discrete harmonic oscillator levels for both repulsive and attractive cases. All of these results are the basis for determining the energy gaps of Cooper pairs arising from two particles interacting in the presence of a Fermi sea consisting of particles immersed in a 2D harmonic trap.
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.
Pairing lies at the heart of superfluidity in fermionic systems. Motivated by recent experiments in mesoscopic Fermi gases, we study up to six fermionic atoms with equal masses and equal populations in two different spin states, confined in a quasi-two-dimensional harmonic trap. We couple a stochastic variational approach with the use of an explicitly correlated Gaussian basis set, which enables us to obtain highly accurate energies and structural properties. Utilising two-dimensional two-body scattering theory with a finite-range Gaussian interaction potential, we tune the effective range to model realistic quasi-two-dimensional scattering. We calculate the excitation spectrum, pair correlation function, and number of pairs as a function of increasing attractive interaction strength. For up to six fermions in the ground state, we find that opposite spin and momentum pairing is maximised well below the Fermi surface in momentum space. By contrast, corresponding experiments on twelve fermions have found that pairing is maximal at the Fermi surface and strongly suppressed beneath [M. Holten et al., Nature 606, 287-291 (2022)]. This suggests that the Fermi sea — which acts to suppress pairing at low momenta via Pauli blocking — emerges in the transition from six to twelve particles.