Specular-Andreev reflection and Andreev interference in an Ising superconductor junction

Abstract

Being resilient to magnetic field, Ising superconductor serves as an exceptional platform for studying the interplay between superconductivity and magnetism. In this Letter, we first explore the transport properties of a two-terminal graphene-Ising superconductor junction where mirage gaps are induced in the superconductor by an exchange field due to magnetic proximity effect. We demonstrate that the chemical potential range of graphene supporting specular-Andreev reflection at the interface is between the two mirage gaps and about twice the Ising spin-orbit coupling strength. This enhances the resilience of observing specular-Andreev reflection against graphene potential fluctuations in experiments. We further study the Andreev interference effect based on a four-terminal junction of which two terminals consist of Ising superconductors in the presence of exchange fields. Due to the finite contribution from the spin-triplet pairing, the interference can be modulated by tuning the relative orientation of the exchange fields in addition to the traditional scheme by changing superconducting phase difference and the chemical potential of the normal region.

Figures (4)

• Figure 1: (a) Schematic of a two-terminal graphene-Ising superconductor junction with translational symmetry along the $y$ direction. (b) Illustration of Andreev reflections at the interface. The mirage gap edges are denoted by $\pm\varepsilon_1$ and $\pm\varepsilon_2$ in addition to the main gap of $2\Delta_{\rm eff}$. The chemical potential of graphene $\mu$ lies within the main gap. Incident electrons from the conduction band (CB) with energies $0<E<\mu$ (blue solid circle) undergo retro-reflection as holes in the same band (blue open circle). Electrons with energies $\mu < E < \Delta_{\rm eff}$ or $\varepsilon_1<E<\varepsilon_2$ (red solid circles) are specularly reflected in the valence band (VB) (red open circles). Normal transmission $T^N$, Andreev reflection coefficient $T^A$ and differential conductance $G/G_0$ at (c) $\mu = 0.2\Delta_0$ and (d) $\mu = 3\Delta_0$, where $G_0$ is the conductance quantum. The effective main gap is $\Delta_{\rm eff} = 0.42\Delta_0$ and the mirage-gap edges are $\pm\varepsilon_1=\pm 5.6\Delta_0$ and $\pm\varepsilon_2=\pm 6.1\Delta_0$.
• Figure 2: (a) Schematic of an Andreev interferometer comprising a central region (boxed area) connected to two Ising superconductors (electrodes $2$ and $4$) and two metallic electrodes (electrodes $1$ and $3$). The central region facilitates both retro-Andreev (dashed blue arrows) and specular-Andreev (solid red arrows) reflections. The interference patterns can be tuned by the relative angle $\theta$ between the exchange fields in the superconductors and the superconducting phase difference $\phi=\phi_4-\phi_2$. The central region and metallic electrodes are modeled using a zigzag graphene nanoribbon. The schematic shows a central graphene region of size $6\times 17$, while the numerical calculations are performed for a region of $80\times 80$. The retro-Andreev and specular-Andreev reflections for the three-terminal configuration [see inset of (b)] are shown at (b) $\mu=0.2\Delta_0$ and (c) $\mu=5.8\Delta_0$. Other parameters are the same as those in Figs. \ref{['fig1']}(c) and (d).
• Figure 3: Interference patterns within the main gap with chemical potential $\mu=0.2\Delta_0$. (a) Interference coefficient $T_{11}^A$ at energy $E=0.1\Delta_0$ [blue star in Fig. \ref{['fig2']}(b)] and (b) interference coefficient $T_{13}^A$ at $E=0.35\Delta_0$ [red star in Fig. \ref{['fig2']}(b)]. Solid lines in (c) and (d) are the line-cuts of the interference patterns for various superconducting phase differences in (a) and (b), respectively. The dashed lines in (c) and (d), scaled by factors of $0.17$ and $0.11$ respectively, correspond to the analytical results given by Eqs. \ref{['T11A']} and \ref{['T13A']}.
• Figure 4: Interference patterns within the mirage gaps with chemical potential $\mu=5.8\Delta_0$. (a) Interference coefficient $T_{11}^A$ at energy $E=5.7\Delta_0$ [blue star in Fig. \ref{['fig2']}(c)] and (b) interference coefficient $T_{13}^A$ at energy $E=6\Delta_0$ [red star in Fig. \ref{['fig2']}(c)]. Solid lines in (c) and (d) are the line-cuts of the interference patterns for various superconducting phase differences in (a) and (b), respectively. The dashed lines in (c) and (d), scaled by factors of $0.3$ and $0.11$, respectively, correspond to $1\pm \cos\phi\cos\theta$ from Eqs. \ref{['T11Am']} and \ref{['T13Am']}.