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arXiv:1807.10914 · Physical review. B./Physical review. B

Optimizing the proximity effect along the BCS side of the BCS-BEC crossover

V. Piselli, Stefano Simonucci, Giancarlo Calvanese Strinati

Published Oct 11, 2018• 8 citations
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Abstract

The proximity effect, which arises at the interface between two fermionic superfluids with different critical temperatures, is examined with a nonlocal (integral) equation whose kernel contains information about the size of Cooper pairs that leak across the interface. This integral approach avoids reference to the boundary conditions at the interface that would be required with a differential approach. The temperature dependence of the pair penetration depth on the normal side of the interface is determined over a wide temperature range also varying the interparticle coupling along the BCS side of the BCS-BEC crossover independently on both sides of the interface. In this way, the size of Cooper pairs evolves from being much larger than (BCS limit) the interparticle distance to being comparable with (unitarity limit, halfway between the BCS and BEC limits) the interparticle distance. Conditions are then found for which the proximity effect is optimized in terms of the extension of the pair penetration depth.

DOI
10.1103/physrevb.98.144508

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The proximity effect, which arises at the interface between two fermionic superfluids with different critical temperatures, is examined with a nonlocal (integral) equation whose kernel contains information about the size of Cooper pairs that leak across the interface. This integral approach avoids reference to the boundary conditions at the interface that would be required with a differential approach. The temperature dependence of the pair penetration depth on the normal side of the interface is determined over a wide temperature range also varying the interparticle coupling along the BCS side of the BCS-BEC crossover independently on both sides of the interface. In this way, the size of Cooper pairs evolves from being much larger than (BCS limit) the interparticle distance to being comparable with (unitarity limit, halfway between the BCS and BEC limits) the interparticle distance. Conditions are then found for which the proximity effect is optimized in terms of the extension of the pair penetration depth.
The Josephson current-phase characteristics, that arise when a supercurrent flows across two fermionic superfluids separated by a potential barrier, can be controlled by varying either the interparticle coupling or the temperature. While the coupling dependence has been addressed in detail both theoretically and experimentally for an attractive Fermi gas undergoing the BCS-BEC crossover, a corresponding study of the temperature dependence of the Josephson characteristics is still lacking in this context. Here, we investigate the combined coupling and temperature dependence of the Josephson characteristics in a systematic way for a wide set of barriers, within ranges of height and width that can be experimentally explored. Our study smoothly connects the two limiting cases, of nonoverlapping composite bosons at low temperature described by the Gross-Piatevskii equation, and of strongly overlapping Cooper pairs near the critical temperature described by the Ginzburg-Landau equation. In this way, we are able to explore several interesting effects related to how the current-phase characteristics evolve along the BCS-BEC crossover as a function of temperature and of barrier shape. These effects include the coherence length outside the barrier and the pair penetration length inside the barrier (which is related to the proximity effect), as well as the temperature evolution of the Landau criterion in the limit of a vanishingly small barrier. A comparison is also presented between the available experimental data for the critical current and our theoretical results over a wide range of couplings along the BCS-BEC crossover.
The properties are considered in detail of a nonlocal (integral) equation for the superconducting gap parameter, which is obtained by a coarse-graining procedure applied to the Bogoliubov-de Gennes (BdG) equations over the whole coupling-versus-temperature phase diagram associated with the superfluid phase. It is found that the limiting size of the coarse-graining procedure, which is dictated by the range of the kernel of this integral equation, corresponds to the size of the Cooper pairs over the whole coupling-versus-temperature phase diagram up to the critical temperature, even when Cooper pairs turn into composite bosons on the BEC side of the BCS-BEC crossover. A practical method is further implemented to solve numerically this integral equation in an efficient way, which is based on a novel algorithm for calculating the Fourier transforms. Application of this method to the case of an isolated vortex, throughout the BCS-BEC crossover and for all temperatures in the superfluid phase, helps clarifying the nature of the length scales associated with a single vortex and the kinds of details that are in practice disposed off by the coarse-graining procedure on the BdG equations.
We calculate the pair correlation function and the order parameter correlation function, which probe, respectively, the intrapair and interpair correlations of a Fermi gas with attractive interparticle interaction, in terms of a diagrammatic approach as a function of coupling throughout the BCS-Bose-Einstein condensation (BEC) crossover and of temperature, both in the superfluid and normal phase across the critical temperature Tc{T}_{c}Tc​. Several physical quantities are obtained from this calculation, including the pair coherence and healing lengths, the Tan's contact, the crossover temperature T∗{T}^{*}T∗ below which interpair correlations begin to build up in the normal phase, and the signature for the disappearance of the underlying Fermi surface which tends to survive in spite of pairing correlations. A connection is also made with recent experimental data on the temperature dependence of the normal coherence length as extracted from the proximity effect measured in high-temperature (cuprate) superconductors.
The strongly correlated regime of the crossover from Bardeen-Cooper-Schrieffer pairing to Bose-Einstein condensation can be realized by diluting a system of two-component fermions with a short-range attractive interaction. We investigate this system via a novel continuous-space-time diagrammatic determinant Monte Carlo method and determine the universal curve Tc/epsilonF for the transition temperature between the normal and the superfluid states as a function of the scattering length with the maximum on the Bose-Einstein condensation side. At unitarity, we confirm that Tc/epsilonF=0.152(7).
We study the stationary Josephson effect for neutral fermions across the BCS-BEC (Bose-Einstein condensate) crossover, by solving numerically the Bogoliubov-de Gennes equations at zero temperature. The Josephson current is found to be considerably enhanced for all barriers at about unitarity. For vanishing barrier, the Josephson critical current approaches the Landau limiting value which, depending on the coupling, is determined by either pair-breaking or sound-mode excitations. In the coupling range from the BCS limit to unitarity, a procedure is proposed to extract the pairing gap from the Landau limiting current.
Most theoretical treatments of inhomogeneous superconductivity/fermionic superfluidity have been based on the Bogoliubov-deGennes equations (or, else, on their various simplified forms), which implement a standard mean-field decoupling in the presence of spatial inhomogeneities. This approach is reliable even at finite temperature for weak interparticle attraction, when the Cooper pair size is much larger than the average interparticle distance (corresponding to the BCS limit of the BCS-BEC crossover). However, it loses accuracy for increasing attraction when the Cooper pair size becomes comparable or even smaller than the average interparticle distance (corresponding to the BEC limit of the BCS-BEC crossover), in particular when finite-temperature effects are considered. In these cases, inclusion of pairing fluctuations beyond mean field is required, a task that turns out to be especially difficult in the presence of inhomogeneities. Here, we implement the inclusion of pairing fluctuations directly on a coarse-graining version of the Bogoliubov-deGennes equations, which makes it simpler and faster to obtain a solution over the whole sector of the temperature-coupling phase diagram of the BCS-BEC crossover in the broken-symmetry phase. We apply this method in the presence of a supercurrent flow, such that problems related to the Josephson effect throughout the BCS-BEC crossover can be addressed under a variety of circumstances. This is relevant in the view of recent experimental data with ultracold Fermi atoms, to which the results of the present approach are shown to favorably compare in the companion paper.