Microscopic theory of superconductor-constriction-superconductor Josephson junctions in a magnetic field

Self-consistent solutions of microscopic Eilenberger theory are presented for a two-dimensional model of a superconducting channel with a geometric constriction. Magnetic fields, external ones, as well as those caused by the supercurrents are included and the relevant equations are solved numerically without further assumptions. Results concerning the influence of temperature, geometric parameters, of $\kappa ={\lambda }_L∕{\xi }_0$, and of external magnetic fields on the Andreev bound states in the weak link and on the current-phase relation are presented. We find that the Andreev bound states within the junction obtain peculiar substructure when a finite supercurrent flows. As long as the London penetration depth is comparable to or bigger than the extension of the constriction, the Josephson effect is independent of $\kappa$. Furthermore, the weak link is very insensitive to external magnetic fields. Features restricted to a self-consistent calculation are discussed.

Figure 1

Two-dimensional model for a ScS Josephson junction consisting of a long channel (width $W=1{\xi }_0$ throughout the paper) with a constriction. The constriction is defined by the parameters $l$ and $w$. Our calculations have been carried out for a section of the channel with length $L=2{\xi }_0$ (dark gray area) and fixed phase difference of the order parameter $\Delta \varphi ={\varphi }_R-{\varphi }_L$. The magnetic field is oriented perpendicular to the image plane (along the $z$ axis).

Figure 2

Josephson effect in a ScS junction with $l=w=0.05{\xi }_0$ and $\kappa =\infty$ at a temperature of $T=0.5T_c$. In (a) and (b), the order parameter amplitude and phase are plotted along the $x$ axis for several values of the phase difference across the junction. The constriction length is indicated with the vertical dashed lines. In (c), we show the corresponding current-phase relation with the Josephson critical current indicated by the dashed lines. The LDOS in the center of the constriction $(x=0,y=0)$ is plotted in (d).

Figure 3

Current flow pattern in the vicinity of the constriction for $l=w=0.05{\xi }_0$, $\kappa =\infty$, and $T=0.5T_c$ at the critical current $(\gamma =0.75\pi )$. For better visibility, we plot only every fourth point out of the grid used in our calculations.

Figure 4

Current-phase relation (a) and LDOS in the center of the constriction (b) for the non-self-consistent S-N-S model given in Eq. (19). The self-consistent current-phase relation from Fig. 2c is also plotted in (a) for comparison. For this figure, we use the same parameters as for Fig. 2.

Figure 5

Current-phase relations for $l=w=0.05{\xi }_0$ and $\kappa =\infty$ at different temperatures are shown in (a) and the corresponding amplitude of the order parameter in the center of the constriction in (b). Points with solid guidelines are the calculated stable branches whereas the dashed lines are a rough sketch of the unstable ones. Inset in (a): temperature dependence of the critical current (points with solid guideline) and the Kulik-Omel’yanchuk (dashed) and the Ambegaokar-Baratoff results (dashed-dotted) for comparison.

Figure 6

Local density of states in the center of the constriction at three different temperatures ($T=0.1T_c$, $0.5T_c$, and $0.9T_c$ from top to bottom) for different values of the phase difference $\gamma$ as indicated in the diagrams. Inset: position of the Andreev bound states at $T=0.9T_c$ as a function of the phase difference $\gamma$ (dots) and $E={\Delta }_{\infty }\left(T\right)\cos (\gamma ∕2)$ for comparison. For this figure, we use the same parameters as for Fig. 5.

Figure 7

Current-phase relation (a) and order parameter amplitude in the center of the constriction (b) for different values of the constriction length $l$ as indicated in the diagrams. For this figure, we use $T=0.5T_c$, $\kappa =\infty$, and $w=0.01{\xi }_0$ and do not consider magnetic fields. The lines are guides for the eyes.

Figure 8

Current-phase relation (a) and order parameter amplitude in the center of the constriction (b) for different values of the constriction width $w$ as indicated. The dashed lines correspond to a channel without constriction. For this figure, we use $T=0.5T_c$, $\kappa =\infty$, and $l=0.01{\xi }_0$ and do not consider magnetic fields. The lines are guides for the eyes.

Figure 9

Current density at the critical current for $\kappa =1$ (a) and $\kappa =0.1$ (b). The dark shading represents low current density, where as the bright areas exhibit high current density. The arrows indicate the direction of the current flow. For small values of $\kappa$, the current is strongly concentrated on the surface of the sample.

Figure 10

Magnetic field distribution $\widehat{\vec{B}}=\widehat{B}{\widehat{e}}_z$ for $\kappa =1$ (a) and $\kappa =0.1$ (b). We multiply the magnetic field $\widehat{B}$ with ${\kappa }^2$ in order to have equal scaling of the vertical axis. The geometry of the sample is indicated with the black lines. For smaller values of $\kappa$, the current is concentrated on the surface of the sample and the magnetic field is suppressed in the interior.

Figure 11

Local density of states in the center of the constriction for two different values of $\kappa$ ($\kappa =0.1$ in the upper panel and $\kappa =1$ in the lower panel). For this figure, we use $T=0.5T_c$ and $l=w=0.05{\xi }_0$. Please note the different scaling of the vertical axis for the two panels.

Figure 12

Current density with applied external magnetic field ($\kappa =1$, ${\widehat{B}}_0=2$) without transport current [$\gamma =0$, (a)] and with critical transport current [$\gamma =0.74\pi$, (b)]. Again, dark shading signifies low current density whereas bright shading signifies high current density. The arrows indicate the direction of the current flow. Without transport currents, a small circulating current appears in the constriction which leads to local antiscreening [white circular arrow in the constriction in (a)]. With an applied phase difference, the total current distribution becomes asymmetric and consists of both screening and transport currents.

Figure 13

Magnetic field distribution $\widehat{\vec{B}}=\widehat{B}{\widehat{e}}_z$ for applied external magnetic field ($\kappa =1$, ${\widehat{B}}_0=2$) without transport current [$\gamma =0$, (a)] and with critical transport current [$\gamma =0.74\pi$, (b)]. The geometry of the sample is indicated with the black lines. Screening currents lead to a reduction of the magnetic field in the interior of the sample. Additional transport currents increase the magnetic field on one side of the sample and decrease it on the other side.

Figure 14

Current-phase relation (a) and order parameter amplitude in the center of the constriction (b) for different values of the external magnetic field ${\widehat{B}}_0$. Inset in (a): dependence of the Josephson critical current on ${\widehat{B}}_0$. Inset in (b): amplitude of the order parameter in the center of the constriction (solid line) and in the center of a long channel without constriction (dashed line) for $\gamma =0$ as a function of the external magnetic field. For the whole figure, we use $T=0.5T_c$, $l=w=0.05{\xi }_0$, and $\kappa =1$. The lines are guides for the eyes.

Figure 15

Local density of states in the center of the constriction for three different values of the external magnetic field (${\widehat{B}}_0=0$, 4, and 8 from bottom to top). For this figure, parameters are set corresponding to Fig. 14. Please note the different scaling of the vertical axis for the three panels.

Figure 16

Temperature dependence of the critical current without magnetic fields ($\kappa =\infty$, ${\widehat{B}}_0=0$, solid guideline) and with external magnetic field ($\kappa =1$, ${\widehat{B}}_0$ as indicated) for $l=w=0.05{\xi }_0$. For comparison, we plot the Kulik-Omelyanchouk result (Ref. 31, dashed line) and the Ambegaokar-Baratoff result (Ref. 32, dashed-dotted line).