Multiple Andreev reflections in weak links of superfluid ${}^3\mathrm{He}\text{−}B$

We calculate the current-pressure characteristics of a ballistic pinhole aperture between two volumes of $B$-phase superfluid ${}^3\mathrm{He}$. The most important mechanism contributing to dissipative currents in weak links of this type is the process of multiple Andreev reflections. At low biases this process is significantly affected by relaxation due to inelastic quasiparticle-quasiparticle collisions. In the numerical calculations, suppression of the superfluid order parameter at surfaces is taken into account self-consistently. When this effect is neglected, the theory may be developed analytically like in the case of $s$-wave superconductors. A comparison with experimental results is presented.

Figure 1

Quasiparticles hitting the wall outside of the constriction are scattered, which leads to a suppression of the superfluid state at distances closer to the wall than ${\xi }_0$ (dashed trajectories). Only quasiparticles hitting the constriction directly are ballistically transmitted and contribute to the current (solid trajectories).

Figure 2

The dc currents $I_0$ for temperatures $T∕T_c=0.4$, 0.5, 0.6, 0.7, 0.8, 0.9 in order of decreasing amplitude. The solid lines include the gap-suppression effects, whereas in the dashed lines it is neglected. The dash-dotted line is the zero-temperature result of Eq. (34), while the straight dashed line corresponds to Eq. (33) at $T∕T_c=0.4$. The results are similar for both parallel and antiparallel ${\widehat{\mathbf{n}}}^{l,r}$’s. Here $F_1^a=0$ and $a=1.6$.

Figure 3

The amplitudes $I_1^S$ and $I_1^C$ for $T∕T_c=0.4$, 0.6, 0.8 in order of decreasing amplitude in the case of parallel ${\widehat{\mathbf{n}}}^{l,r}$’s. The solid and dashed lines are for $I_1^S$ with and without gap suppression, respectively, and the dash-dotted and dotted lines are the corresponding values for $I_1^C$. Here $F_1^a=0$ and $a=1.6$.

Figure 4

Same as Fig. 3 but for $I_2^S$ and $I_2^C$.

Figure 5

Same as Fig. 3 but for antiparallel ${\widehat{\mathbf{n}}}^{l,r}$’s.

Figure 6

Same as Fig. 4 but for $I_2^S$ and $I_2^C$.

Figure 7

Local density of states inside the pinhole for ${\widehat{k}}_z=0.93$ and $-{\widehat{\mathbf{n}}}^l={\widehat{\mathbf{n}}}^r=\widehat{\mathbf{z}}$ when gap suppression at a diffusive wall is included. Here a large, arbitrary value $a\approx 75$ was chosen for purposes of illustration, in order to make the width $\sim \hslash \Gamma$ of the bound-state peaks better observable. Among the peaks, those corresponding to the $\sigma =+1$ $(\sigma =-1)$ spin branch are the ones apparently shifted toward smaller (larger) $\varphi$. Note that the slope of the bound states is not so steep as for Eq. (26). Also, for given phase difference and spin band, more than one bound state can exist simultaneously.

Figure 8

Comparison of the data ($+$ signs) from Ref. 12 to the pinhole theory (solid lines) for a diffusive surface and $a\approx 0.27$ at indicated temperatures.

Figure 9

Comparison of the $L$ state data ($+$ signs) from Ref. 13 to the pinhole theory (solid lines) for a diffusive surface and $a\approx 1.6$ at indicated temperatures.