Even-odd flux quanta effect in the Fraunhofer oscillations of an edge-channel Josephson junction

We calculate the beating of $h/2e$ and $h/e$ periodic oscillations of the flux-dependent critical supercurrent $I_c\left(\Phi \right)$ through a quantum spin-Hall insulator between two superconducting electrodes. A conducting pathway along the superconductor connects the helical edge channels via a nonhelical channel, allowing an electron incident on the superconductor along one edge to be Andreev reflected along the opposite edge. In the limit of small Andreev reflection probability the resulting even-odd effect is described by $I_c\propto |cos(e\Phi /\hslash )+f|$, with $\left|f\right|\ll 1$ proportional to the probability for phase-coherent interedge transmission. Because the sign of $f$ depends on microscopic details, a sample-dependent inversion of the alternation of large and small peaks is a distinctive feature of the beating mechanism for the even-odd effect.

Figure 1

Beating mechanism for the even-odd effect in the Fraunhofer oscillations. For uncoupled edges the flux periodicity is $h/2e$, corresponding to the transfer of a charge-$2e$ Cooper pair along the left or right edge channel [blue (red) hatched strips]. The edge channels are coupled by a conducting path along the NS interface, allowing for a circulating loop of charge $\pm e$ with $h/e$ flux periodicity. The circulating loop may be partly $e$ type (red lines) and partly $h$ type (blue), as in panel (a), or it may be entirely of one charge type [entirely $e$, as in panel (b), or entirely $h$]. Both loops contribute to the even-odd effect, but panel (a) dominates when the Andreev reflection probability $\Gamma$ is small. [It is of order $\Gamma$, while panel (b) is of order ${\Gamma }^2$.]

Figure 2

Josephson junction in a current-biased circuit [panel (a)], to study the dependence of the critical current $I_c$ on the magnetic flux $\Phi$ enclosed by a circulating edge channel [panel (b)]. The network model of the Josephson junction is illustrated in panel (c). Helical modes (red, amplitudes $a_{\uparrow },a_{\downarrow }$) and nonhelical modes (blue, amplitudes $b_{\uparrow },b_{\downarrow }$) are coupled at four nodes by a scattering matrix $s_n$, relating incoming and outgoing amplitudes.

Figure 3

Even-odd effect in the Fraunhofer oscillations of the critical current due to the beating of $h/e$ and $h/2e$ oscillations. The curves are calculated with spin mixing from Eq. (18) [solid lines, dominated by the path of Fig. 1] and without spin mixing from Eq. (19) [dashed lines, dominated by the path of Fig. 1].

Figure 4

Fraunhofer oscillations for the experimentally relevant parameters given in the text, calculated from Eq. (8) for three different temperatures.

Figure 5

The solid curve is the $T=20\mathrm{mK}$ critical current of Fig. 4, without phase shifts at the scattering nodes, while the dashed curve shows the inverted even-odd effect for ${\psi }_1^{\prime }+{\psi }_3^{\prime }=\pi$ (and all other phase shifts kept at zero).