SNS Junctions along the BCS-BEC Crossover

Abstract

We present a theory of SNS junctions, a normal metal sandwiched between two superconductors, along the crossover from the BCS to the BEC regime. We calculate the Josephson current as a function of the chemical potential relative to the band edge in the superconducting region, $μ_S$, where the BEC phase is indicated by $μ_S <0$. The chemical potential relative to the band edge in the normal metal, $μ_N$, allows us to tune the junction between the SNS case ($μ_N>0$) and the SIS case, where the superconductors are separated by a tunneling barrier. We find that there are Andreev levels in the BEC regime, as long as there is sufficient density of states in the normal region, i.e. when $μ_N>Δ$, where $Δ$ is the amplitude of the superconducting order parameter. For 1D SNS junctions, we find the Josephson current $I_S$ carried by these Andreev levels to be a function of the ratio $Δ/Δ_d$, where $Δ_d$ is the Andreev level spacing. At zero temperature, the Josephson current has a maximum on the BCS side of the transition where $Δ$ is maximal. At finite temperature, however, we find that the maximum moves to the BEC side of the crossover. We identify the mechanism for this phenomenon to be the decrease in the number of Andreev levels at the BCS-BEC crossover, accompanied by an increase in excitation energy to the unoccupied levels, making it less likely that these states are thermally occupied. Thereby, at finite temperature, the Josephson current is more strongly reduced on the BCS side of the crossover, resulting in a maximal Josephson current at the BCS-BEC crossover.

Figures (6)

• Figure 1: a) SNS-junction, with a normal region (N) sandwiched between two superconducting leads (S). b) electron and hole dispersions in each region. Blue lines: electron and hole energies relative to the chemical potential $\mu$, $E,-E$. Red arrows: Andreev Scattering of electron with momentum $k_e$ to hole with momentum $-k_h$ and back, forming a clockwise standing wave, an Andreev level. The anticlockwise standing wave, from $-k_e$ to $k_h$ and back, forms another Andreev level (not shown). Red, blue solid lines: band edge in the superconducting, normal region, $E_{CS},E_{CN}$, respectively.
• Figure 2: Schematic density of states of a) BCS-N-BCS junction, b) BCS-I-BCS junction with tunneling barrier $E_{CN}-\mu$.
• Figure 3: Schematic density of states of a) BEC-N-BEC junction with $\mu_N = \mu-E_{CN} > \Delta_G$, b) BEC-N-BEC junction with $\mu_N = \mu-E_{CN} < \Delta_G.$
• Figure 4: Andreev levels, Eq. (\ref{['enplus']}), as function of phase difference $\Delta \phi$ and ratio of order parameter and level spacing $\Delta/\Delta_d= d/(\xi \pi)$.
• Figure 5: a) The universal function $g(\Delta/\Delta_d,\Delta \phi =\pi,T=0)$. b) Order parameter $\Delta$ of a 1D superconductor as function of $\mu_S$ in units of $t$, and Josephson current $I_s$ through an SNS junction between 1D superconductors at $T=0K$ at $\Delta \phi = \pi$ as function of $\mu_S$ in units of $t$. We set $U= 4 \pi t/10,$$\mu_N= 0.5t$ and $\Delta_d/t =1/(2\sqrt{2}).$