Superconducting tunnel junctions with layered superconductors

Abstract

The superconducting tunneling effect in heterostructures, describing the process where single electrons or Cooper pairs tunnel through the barrier, can always play a significant role in understanding the phase coherence and pairing mechanisms in superconductors. Taking advantage of the easy cleavage to atomically-thin monolayer structure of layered superconductors and resulting quantum confinement of electrons or Cooper pairs at two-dimensional limit, van der Waals superconducting materials hosting superconducting order in monolayers or heterostructures can exhibit extensive emergent phenomena associated with quantum phase transitions of vortex and anti-vortex pairs. Examples of superconducting tunnel junctions (STJs) based on layered superconductors have been demonstrated to achieve novel phenomena, including Andreev bound states, Majorana bound states and 0/π-phase junctions. Since the characteristic parameters of quasiparticle tunneling through the barrier are directly associated with the energy gap values of superconductors, such critical parameter can be obtained within the STJ device geometry, which helps us understand and control the pairing states and emerging phenomena in superconductors. In this review, from the perspective of STJs with single electron tunneling and Cooper pair tunneling, we discuss Andreev reflection, Majorana bound states, photon-induced tunneling effects, non-reciprocal transport and superconducting diode phenomena, as well as prospects for layered-superconductor-based STJs.

1 Introduction

Superconducting tunnel junctions (STJs), being an efficient approach to investigating the quantum tunneling processes of quasiparticles in superconductors, can serve as an extraordinary platform for exploring novel electronic states and strong correlation phenomena of condensed matters [1–7]. The technological appeal of STJs is the exceptional sensitivity, precision, and speed in detecting and manipulating quantum-level phenomena. Examples of STJs have been demonstrated in understanding Majorana zero modes [8–20], exploring Andreev bound states [21–23], and probing superconducting diode physics [24–35], as well as developing superconducting single-photon detectors [7, 36–47]. Interestingly, the cleavable layered superconductors with reduced dimensionality and strong electron correlation can possess novel mechanisms beyond those in conventional Bardeen-Cooper-Schrieffer (BCS) superconductors [48], including p- or d-wave pairing [49–55], Berezinskii–Kosterlitz–Thouless transition [56–60], Ising pairing [61–63], and flat-band superconductivity [64–67], as well as quantum Griffiths singularity [68–71]. Therefore, STJs with layered superconductors are of great significance for understanding novel superconducting pairing mechanisms and achieving phase-sensitive quantum phenomena in strongly correlated systems.

In superconductor/barrier/superconductor sandwiched STJ structures, the quasiparticles undergoing the tunneling process can be classified into the following two types: single electrons (Bogoliubov quasiparticles to be precise) or Cooper pairs, as shown in Fig. 1a, b. Generally, a quantitative criterion for distinguishing the two basic quasiparticle tunneling processes can be expressed by comparing the tunnel barrier thickness d and the coherence length ξ of Cooper pairs along the tunneling direction (Fig. 1c) [72]. For $d>\xi$, only single electron tunneling occurs (Fig. 1a), while for $\xi >d$, Cooper pair tunneling dominates the entire tunneling process (Fig. 1b, such an STJ is known as a Josephson junction). For the former case of $d>\xi$ with a relatively thicker tunnel barrier, since the superconducting order parameter exponentially decreases within the barrier, the coherence between two superconductors through such a thick barrier is ignorable. Therefore, the direct tunneling of Cooper pairs through the tunnel barrier is forbidden, and a single electron tunneling process is expected (Fig. 1a). For the latter case with $\xi >d$, when the thickness of the sandwiched tunnel barrier is small enough, the order parameters of the two superconductors can coherently interact with each other, and such thin tunnel barriers will always cause a phase shift between two superconducting order parameters, so Cooper pairs are expected to directly tunnel through the barrier (Fig. 1b), serving as the characteristics of Josephson junctions [73].

Figure 1

Microscopic mechanisms of quasiparticle tunneling and the corresponding characteristics of STJs with layered superconductors. (a) Schematic illustration of STJ structure with single electron tunneling. (b) Schematic illustration of STJ structure with Cooper pair tunneling. (c) The type of quasiparticle tunneling depends on the relation between the superconducting coherence length ξ (upper panel) and the characteristic thickness d of the barrier (lower panel). The condition of $\xi < d$ can normally induce the single electron tunnel process (d), while the condition of $d<\xi$ can induce the Cooper pair tunnel process (e). The band alignment features between the two superconductors for the two types of quasiparticle tunneling and the corresponding I-V characteristics are schematically shown in (d) and (e). An increase in the tunnel current will occur when the bias voltage is $( \Delta _{1} + \Delta _{2} )/e$ for a single electron tunnel, and a non-zero supercurrent $I_{c }$ at zero bias voltage can be found for Josephson junctions. Note that a hysteresis loop can typically be observed in the I-V characteristics of the Cooper pair tunnel. The upper panels in (d) and (e) are adapted from Ref. [72]

Experimentally, electronic transport measurements (for example, current-voltage I-V characteristics) are generally used to distinguish the single electron or Cooper pair tunneling processes [73–76]. Based on the definition of the superconducting gaps as $\Delta _{1}$ and $\Delta _{2}$ for the left and right superconductors, we can obtain the following understanding. (1) For the case of single electron tunneling where $\xi < d$, when the bias potential exceeds $( \Delta _{1} + \Delta _{2} )/e$ (e is the elementary charge), electrons below the superconducting gap from the left superconductor can tunnel through the barrier, entering the right superconductor, as demonstrated in the upper panel of Fig. 1d. However, the electrons can hardly tunnel through the barrier layer if the potential bias is smaller than $( \Delta _{1} + \Delta _{2} )/e$ due to the lack of density of states at the Fermi level. As a result, the tunnel current is small for such a situation, as shown in the bottom panel of Fig. 1d. (2) For the case of Cooper pair tunneling where $\xi >d$, even with zero bias voltage, a non-zero supercurrent through the Josephson junction can occur because of the superfluidity of Cooper pairs, showing as the evidence for the intrinsic difference between single-electron STJs and Josephson junctions. Further increasing the electrical current over the critical supercurrent $I_{c }$ of the Josephson junction, the non-zero bias voltage between the two superconductors can be detected due to single electron tunneling (Fig. 1e).

The STJ device geometry using atomically-thin layered superconductors or their heterostructures can host the following notable advantages for understanding the underlying pairing mechanism in superconductors and quantum phase transitions therein. First, the quasiparticle tunneling feature within STJs can directly provide detailed information about the density of states of the superconductor, in particular for the superconducting gap values and the phase coherence between superconducting order parameters [77, 78]. Second, the “transfer and stack” technique to form high-quality van der Waals (vdW) heterostructures [79–81] can directly provide an atomically-flat and high-quality interface between the superconductor and the tunnel barrier in an STJ [78, 82, 83]. Third, unlike the conventional sandwiched STJ structure where an insulating barrier is necessary, an STJ can also be formed by stacking two exfoliated layered superconductors separated by an insulating vdW gap between them [52, 84], which provides a convenient approach to understanding the details of superconducting gap states. Fourth, the superconducting behavior of layered superconductors can be maintained even in the two-dimensional (2D) limit [51, 62, 85], so the phase-sensitive measurement based on STJ structures can be applied to determine the pairing symmetry therein [52].

This review focuses on recent forefront research involving STJs based on layered superconductors. The main topics we aim to cover here include Andreev scattering, Majorana bound states, single photon detection, 0/π-phase junctions and superconducting diodes [88–95], as shown in the content summary in Fig. 2. We also discuss several important topics, including the non-reciprocal transport phenomena by symmetry engineering and the Majorana bound states for topological quantum computation by the proximity effect in STJs.

Figure 2

Overview of the structure, physical mechanisms and functionalities of STJs with layered superconductors. Two typical structures of STJs can normally be fabricated: out-of-plane STJs and in-plane STJs. Regarding the quasiparticle tunnel in STJs, different physical processes can be studied in the two types of STJs. For the device application of STJs, a scanning superconducting quantum interference device, amplifier, photodetector, as well as other functionalities can be achieved. Pictures from top to bottom in panel (a) are adapted from Ref. [86], Ref. [24] and Ref. [87]. Pictures from left to right in panel (b) are adapted from Ref. [88], Ref. [89] and Ref. [90]. Pictures from left to right in panel (c) are adapted from Ref. [91], Ref. [92], Ref. [93] and Ref. [94]. Pictures from top to bottom in panel (d) are adapted from Ref. [95], Ref. [96] and Ref. [42]

2 Geometry of superconducting tunnel junctions

Since the coherence length ξ of layered superconductors is often be highly anisotropic along the in-plane and out-of-plane directions [62, 99], the characteristic length d of the tunnel barrier (for example, d represents the barrier thickness for the out-of-plane STJs) should be carefully selected to realize STJs with different quasiparticle tunneling. The device geometry of the out-of-plane or in-plane STJs and their distinct anisotropic transport properties are pivotal in understanding the symmetry breaking-induced nonlinear and non-reciprocal electronic transport phenomena therein.

In out-of-plane STJs where electron tunneling occurs perpendicular to the interface between the superconducting nanoflake and the barrier (Fig. 3), the selection of tunnel barrier thickness typically depends on the out-of-plane ξ value of layered superconductors [74, 87, 100, 101]. Regarding the fabrication process of out-of-plane STJs, the “bottom-up” method of multiple-layered vdW superconducting films is a widely used technique to form STJs with clean and flat interfaces. For example, large area and homogenous STJ devices can be fabricated by using metal evaporation (Al/InGaAs/Al, Fig. 3a) [97], epitaxial growth technique (La_2-xSr_xCuO₄/La₂CuO₄/La_2-xSr_xCuO₄, Fig. 3b) [98], or chemical vapor deposition (NbSe₂/MoSe₂/NbSe₂, Fig. 3c) [87]. Considering that the atomically-thin superconducting nanoflakes can also be achieved via the “top-down” exfoliation technique [81, 102–104], stacking nanoflakes together can serve as an effective way to fabricate the STJ sandwiched structures. Since the large vdW gap between the stacked nanoflakes can directly serve as the tunnel barrier in STJ sandwiched structure, the stacked NbSe₂/NbSe₂ (Fig. 3d) or Bi₂Sr₂CaCu₂O_8 + δ/Bi₂Sr₂CaCu₂O_8 + δ homojunctions with the vdW gap tunnel barrier [52, 84] can function as typical STJ devices. Also, the NbSe₂/Nb₃Br₈/NbSe₂ STJ structure with inversion-symmetry-breaking tunnel barrier Nb₃Br₈ (Fig. 3e) can be used to form superconducting diodes and realize non-reciprocal supercurrents [24]. Such a stacking technique can offer an extra control degree of freedom to modulate the properties of STJs by changing the thickness or crystal symmetry of the barrier layer.

Figure 3

Structure of the out-of-plane STJ with layered superconductors based on “bottom-up” (a)–(c) and “top-down” (d), (e) approaches. (a) Schematic illustration of the Al/InGaAs/Al out-of-plane STJ prepared by the metal evaporation method. (b) LSCO/LCO/LSCO out-of-plane STJ prepared by the epitaxy method, characterized by transmission electron microscopy. (c) Chemical vapor deposition method for layered superconductor NbSe₂ to form an out-of-plane STJ. Wafer-scale growth of the heterojunction can be achieved, as confirmed by the optical image. (d) Out-of-plane STJ achieved by stacking two NbSe₂ nanoflakes, investigated by transmission electron microscopy. The vdW gap between the two nanoflakes (light gray region in the middle) can serve as the tunnel barrier. (e) Schematic illustration (upper panel) and optical image (lower panel) of the out-of-plane STJ fabricated by transfer and stacking of the NbSe₂/Nb₃Br₈/NbSe₂ sandwich structure. Panel (a) is adapted from Ref. [97]. Panel (b) is adapted from Ref. [98]. Panel (c) is adapted from Ref. [87]. Panel (d) is adapted from Ref. [84]. Panel (e) is adapted from Ref. [24]

In in-plane STJs where the superconductors and the barrier layer are on the same plane of substrate surface, a designed weak link region between two layered superconductors (named as “Dayhem bridge”) can directly serve as the tunnel barrier for quasiparticle tunneling processes [86, 105, 106, 108–115], as demonstrated in Fig. 4a-c. Considering that the ξ value along the in-plane direction is normally much larger than that along the out-of-plane direction [98, 116–120], two advantages of such in-plane STJs should be addressed here: On the one hand, it is easy to achieve quantum interference among multiple STJs, which are the core component of superconducting quantum interference devices for precise detection of magnetic fields [121–124]. On the other hand, in-plane STJs can be easily incorporated in a circuit, making them excellent candidates for applications in topological quantum computation and quantum information processing [125–130]. For example, as shown in Fig. 4d, the nanoscale STJs fabricated by shadow evaporation technique can serve as qubits for superconducting quantum computation [107, 131, 132]. Remarkably, only based on such a planar structure (Fig. 4e), a tunable superconducting qubit called “Xmon” of quantum computation can reach an excellent energy coherence time exceeding 40 μs [91].

Figure 4

Structure of the in-plane STJ with layered superconductors. (a) In-plane STJ formed at the LaAlO₃/SrTiO₃ interface achieved by the epitaxy method. The weak link region is highlighted as “WL” in the atomic force microscope image (lower panel). (b) In-plane STJ based on NbSe₂ fabricated by the reactive ion etching technique. Two weak link regions can be clearly observed in the scanning electron microscopy (SEM) image. (c) Schematic of an in-plane STJ based on YBa₂Cu₃O_7 − δ obtained by a focused helium ion beam. (d) SEM image of a d.c.-superconducting quantum interference device consisting of two in-plane Josephson junctions for quantum computing. (e) The planar Xmon qubit based on in-plane STJs. The qubit is capacitively coupled to a quarter wave readout resonator (top), a quantum bus resonator (right), and an XY control line (left) and inductively coupled to a Z control line (bottom), where the Xmon arm length is L. The upper right and lower right panels show the electrical circuit of the qubit and the fine structure of the in-plane STJ. Panel (a) is adapted from Ref. [86]. Panel (b) is adapted from Ref. [105]. Panel (c) is adapted from Ref. [106]. Panel (d) is adapted from Ref. [107]. Panel (e) is adapted from Ref. [91]

3 STJ with single electron tunneling

Normally, whether the single electrons can tunnel through the barrier or not depends on the energy gap values of the superconductors on both sides, and also on their relative position in the density of state. Since such a tunneling process can be clearly described by the typical tunneling process through a potential barrier with finite height in quantum mechanics, the detailed features of band structures and the superconducting gap can be reflected within the single electron tunneling process [74, 75, 101, 133–137], as shown in Fig. 1d. Interestingly, once we change the barrier layer to those semiconductors with strong spin-orbital coupling, 2D topological insulators, or Dirac/Weyl semimetals, such kind of electron tunneling process can help us investigate the topologically non-trivial phenomena, including the p-wave pairing mechanism, Majorana bound states, and unconventional Josephson effect [138–147]. In the subsequent subsections, we will explore novel quantum phenomena based on STJs with single electron tunneling.

3.1 Probing details of the superconducting gap with STJ devices

Generally, in a superconductor-insulator-superconductor structure, the Δ values for the two superconductors ($\Delta = \Delta _{1} + \Delta _{2}$) hinder the direct determination of each gap value of the two superconductors (Fig. 1d). While if we change the superconductor on one side to a normal metal (i.e., $\Delta _{2} =0$) to form superconductor-insulator-metal (S-I-N) structure, the gap value of the other superconductor can be precisely determined in such an STJ structure. Since there is only one superconductor in the S-I-N structure, the turning point in the I-V or $\mathrm{d} I/ \mathrm{d} V$-V characteristics of the S-I-N structure corresponds to $\Delta / e= \Delta _{1} /e$ instead of $( \Delta _{1} + \Delta _{2} )/e$, which can be well-explained by the BCS theory for electron pairing [74, 101, 134].

Based on this method, the Δ value of the layered superconductor NbSe₂ and the evolution of Δ with external magnetic fields can be understood in a NbSe₂/MoS₂/Au structure with a MoS₂ tunnel barrier, as shown in Fig. 5a, b [78]. In such a structure, the researchers demonstrated that vdW tunnel barriers could sustain stable currents with strong suppression of subgap tunneling and offer the opportunity to distinguish the two intrinsic superconducting gaps in such a multiband superconductor. Based on such an S-I-N structure, the researchers clearly confirmed the presence of two superconducting order parameters and found that the larger superconducting gap decreases monotonically with thickness and critical temperature $T_{c }$. The left panel of Fig. 5a presents the evolution of the $\mathrm{d} I/ \mathrm{d} V$ curve of bulk-form NbSe₂ under an in-plane magnetic field, and the shortening of the conductance peaks under a higher field can be observed. The experimental results can be well fitted by the Suhl-Schopohl-McMillan model including the diffusive de-pairing process [150, 151], as shown in Fig. 5b. Such a technique can be generalized to other materials that have hitherto not been investigated using tunneling spectroscopy.

Figure 5

Probing the superconducting gap values of layered superconductors based on the STJ structure. (a) Optical image and the in-plane field dependence of tunneling conductance spectra of an out-of-plane STJ of 2H-NbSe₂, measured at 70 mK. (b) Thickness dependence of the tunneling conductance for 3-layer, 4-layer and bulk-form NbSe₂ measured at 70 mK. Black curves are the fitting results based on Suhl-Schopohl-McMillan theory. (c) Scanning tunneling spectrum of the 2H-NbS₂ nanoflake. The inset figure shows the differential of the bias voltage-dependent local density of states. Two peaks highlighted by black and gray arrows can be observed in the spectrum, indicating the two-band nature of superconducting NbS₂. (d) Evolution of fitted Δ values with temperature. (e) The first-principles calculations of the superconducting gap values of 2H-NbS₂. The calculation results agree well with the experimental results. (f) Schematic illustration of electronic transport measurement in a diamond anvil cell. (g) Tunneling conductance of an out-of-plane STJ formed by superconducting GeP and multilayer graphene in a diamond anvil cell measured at 1.5 K. Curves are fitting results with different Z parameters based on the Blonder-Tinkham-Klapwijk model (marked as “BTK” in the figure). Panels (a) and (b) are adapted from Ref. [78]. Panels (c) and (d) are adapted from Ref. [148]. Panel (e) is adapted from Ref. [149]. Panels (f) and (g) are adapted from Ref. [77]

To capture the energy gap details in the band structures of superconductors, scanning tunneling microscopy (STM) measurements are carried out, where the testing geometry can be regarded as a generalized S-I-N STJ structure with vacuum tunneling barrier. Take superconducting 2H-NbS₂ as an example [148], a peculiar superconducting density of states with two well-defined features at 0.97 and 0.53 meV can be observed, which are above and below the value for the superconducting gap expected from the single band s-wave BCS model ($\Delta =1.76 k_{\mathrm{B}} T_{c } =0.9 \mathrm{meV}$, where $k_{\mathrm{B}}$ is Boltzmann constant). Note that these two features in the differential of density of states can be clearly confirmed (highlighted by the black and gray arrows in the inset of Fig. 5c) in the differential conductance curve $\mathrm{d} I/ \mathrm{d} V$-V. Correspondingly, the temperature dependence of the two subgap values are shown in Fig. 5d, and such results are in good accordance with the first-principles calculations [149] in Fig. 5e. Unlike transport or specific-heat experimental methods that can only provide secondary evidence of multiband superconductivity [119, 152, 153], such a two-peak feature obtained by STM can directly indicate the existence of two superconducting subgaps and serve as smoking-gun evidence of multigap superconductivity and orbital-selective 2D superconductivity in 2H-NbS₂. In unconventional multiband superconducting systems such as iron-based and nickelate superconductors, STM has been applied to successfully measure quasiparticle interference patterns based on the tunnelling spectra, aiding in understanding the pairing mechanism therein [154].

More interestingly, for the superconducting parameters and pairing mechanisms of pressure-induced superconductivity, which is hard to detect with the STM technique mentioned above within the diamond anvil cell chamber [155], an STJ could provide a possibility to probe the density of states and the superconducting gap of pressure-induced superconductors. Figure 5f shows a schematic illustration of the electronic transport measurement geometry for GeP/graphene/Au STJs inside a diamond anvil cell [77]. By stacking multilayer graphene on top of the studied material GeP, an out-of-plane STJ can be obtained at high pressure after GeP has gone through an insulator-to-superconductor transition. The bias voltage dependence of the tunneling conductance spectra can thus be obtained and well fitted by the Blonder-Tinkham-Klapwijk model for an Andreev reflection process, as shown in Fig. 5g (to be further discussed in Sect. 3.3). Therefore, the Δ value can be precisely determined by the fitting parameters, solving the problem of probing the superconducting gap value in high-pressure-induced superconductors.

3.2 Superconducting proximity effect in the STJ

The superconducting proximity effect describes the phenomenon that, when a metal or semiconductor is in contact with a superconductor and when the thickness of the non-superconducting material is less than the coherence length ξ of the superconductor, the initially non-superconducting material will also become a superconductor due to non-zero superconducting order parameters therein. In the induced superconducting region, the coherence length and the wavefunction of the Cooper pair distribution can be controlled by the barrier thickness, the type of barrier materials, and the $T_{c }$ values of superconductors, as well as the strength of the applied magnetic fields. By adjusting these parameters in STJs, one can effectively modulate superconducting states for designed coherence length, specific transmission coefficients and Andreev bound states [9, 156–161]. Specifically, in a superconductor-normal metal (S-N) junction, Cooper pairs can emerge within the normal metal component, as schematically illustrated in Fig. 6a. This type of junction shows unusual conductance behavior and can provide an excellent platform for creating intrinsically shunted junctions [162]. More importantly, if two S-N junctions are connected together to form the S-N-S junction, a wide range of intriguing phenomena, including the emergence of supercurrent in the normal metal region and the modulation of conductance in Andreev interferometers by magnetic flux, can be realized [163, 164]. We will summarize the experimental phenomena of these structures in the following paragraphs.

Figure 6

Superconductor proximity effect in STJs with normal metals and semiconductors. (a) Schematic illustration of the proximity-induced superconductivity in an S-N heterostructure. (b) Schematic side view of the NbSe₂/bilayer graphene junction. The differential junction resistance $R_{ns} = \mathrm{d}V_{{ac}}/\mathrm{d}I_{{ac}}$ is measured by passing a d.c. and a small a.c. current from electrodes 4 to 1 and measuring the a.c. voltage between electrodes 11 and 3. (c) Normalized conductance $G_{{ns}}$ ($I_{{dc}}$) with a back-gate $V_{{g}} = -40$ V at different temperatures. (d) $R_{{ns}}$ ($I_{{dc}}$) for $V_{{g}}=+40$ V at different temperatures. The carrier types are holes and electrons for (c) and (d), respectively. (e) $I_{{dc}}-T$ relations of the circles, triangles and diamonds marked in (c) and (d). The black curve is a fit to the BCS gap model: $I_{{dc}} = \tanh (1.74 \sqrt{\frac{T_{{c}}}{T} -1} )$, where $T_{{c}}= 6.8$ K. (f) Illustrations of the two gaps and two Andreev reflection processes in the NbSe₂/bilayer graphene junction. (g) Upper: Schematic of a three-terminal hybrid device and the measurement circuit. Middle: Gate-dependent gap values of superconductors $\Delta _{{sc}}$ and induced superconductivity in semiconductors $\Delta _{{i}}$. Lower: Calculated nonlocal slope at zero bias for $g_{{RL}}$ (purple) and $g_{{LR}}$ (orange). Panels (b)–(f) are adapted from Ref. [165]. Panel (g) is adapted from Ref. [166]

Ultra-transparent S-N junctions based on NbSe₂ and bilayer graphene were used to demonstrate a proximity-induced superconducting state (Fig. 6b) [165]. As shown in Fig. 6c, the normalized conductance $G_{ns}$ clearly displays a distinct feature characterized by a “dome plus peak” structure, in which the onset of the dome (blue triangles) matches well with the superconducting gap of NbSe₂. Since the enhancement in the $G_{ns}$ peak observed in Fig. 6d provides strong evidence of an exceptionally transparent interface, the BCS superconducting gap $\Delta (T)$ model can describe the temperature dependence of the triangle, diamond, and circle points shown in Fig. 6e very well, indicating that all three thresholds are directly associated with the superconducting gap ($\Delta _{0}$) in NbSe₂ based on proximity effects. The induced gap $\Delta _{\mathrm{BLG}}$ that arose in the bilayer graphene region directly in contact with NbSe₂ is associated with the proximity effect (Fig. 6f). This study establishes the understanding of an all-vdW, high-performance superconducting junction and opens up possibilities to engineer emergent quantum phenomena.

The superconducting proximity effect in semiconductor-superconductor nanowires is expected to generate an induced superconducting gap in the semiconductor. The magnitude of this induced gap, together with semiconductor properties such as spin-orbit coupling and the Landé g-factor, depends on the coupling between the neighboring materials [167, 168]. For example, the InSb/Al/Pt hybrid structure can be tuned by applying a gate voltage such that the semiconductor and superconductor are strongly coupled [166]. The coupling can be suppressed, which leads to a strong reduction in the induced gap and critical magnetic field, and the closing and reopening of the induced gap can be observed at the crossover between the strong-coupling and weak-coupling regimes. According to the relation between the superconducting gap and gate voltage ($V_{SG}$) shown in the lower panel of Fig. 6g, the induced superconducting gap ($\Delta _{i}$) approaches 0 above $V_{SG}> 6$ V, confirming that the hybrid structure becomes gapless. Bias voltages, $V_{L }$ and $V_{R }$, were individually applied to the left and right leads, and the differential conductance $g_{ij}= \mathrm{d}I_{i}/\mathrm{d}V_{j}$, with $i, j= L$, R was obtained using standard lock-in techniques shown in the upper panel of Fig. 6g. When the induced superconductivity ($V_{SG}< 6$ V) exists in the hybrid, the parameters $S_{RL}$ and $S_{LR}$ (defined as $S_{ij} = \frac{\mathrm{d}^{2} I_{i}}{\mathrm{d} V_{j}^{2}} | V_{j} =0$), remain extremely close to 0, suggesting strong coupling between the superconductor and the semiconductor. This hybrid structure offers a platform for tuning the proximitized superconducting state, opening up opportunities for mapping the local charge of Andreev bound states [169], investigating the quasiparticle wavefunction composition [170, 171] and better understanding topological phase transition [172].

3.3 Andreev bound states in STJs with layered superconductors

Andreev scattering in STJ structures describes the process in which an incoming particle-like excitation can convert to an outgoing hole-like excitation at the interface between a superconductor and a non-superconducting material [175]. Two kinds of Bogoliubov quasiparticles are involved in such a process: particle-like excitations characterized by a group velocity $\boldsymbol{v}_{p }$ aligned with the direction of the particle momentum p, $\boldsymbol{v}_{p } \cdot \boldsymbol{p} > 0$, and hole-like excitations with reversed group velocity, $\boldsymbol{v}_{p } \cdot \boldsymbol{p} < 0$ [23]. Different from the specular electron reflection at a metal-insulator interface (Fig. 7a), the reflected hole is retro-reflected in Andreev reflection (Fig. 7b). Andreev scattering is realized by the transport of charge and momentum by Cooper pairs in compliance with the conditions of charge neutrality and momentum conservation. Importantly, multiple Andreev scattering events occur when there are pronounced spatial variations in superconducting gap $\Delta (\boldsymbol{r})$ (typically at the superconductor-metal interface or the cores of quantized vortices), leading to the formation of Andreev bound states. These bound states are localized near the region of significant spatial variations in $\Delta (\boldsymbol{r})$, as shown in Fig. 7c. A corresponding schematic illustration of the Andreev reflection process in an STJ device is shown in Fig. 7d.

Figure 7

Investigation of Andreev bound states in STJs with layered superconductors. (a)-(c) Specular reflection of an electron at a metal-insulator boundary (a), N-S boundary (b), and the Andreev bound state confined within an S-N-S sandwich (c). (d) Schematic illustration of the Andreev reflection process in an STJ device geometry. (e)-(g) Theoretical calculation of the universal conductance scaling of Andreev reflections. (e) A model of a superconductor-semiconductor junction. (f) Possible physical processes that can appear in such a structure. (g) Scaling dimension and amplitude (inset) of the operator for dimensionless parameters $r = 0.1$ (upper panel) and 0.45 (lower panel). (h)-(k) In-plane magnetic field-tunable Andreev bound states based on an STJ with a layered superconductor. (h) Structure of the heterojunction. (i) Band structure of such a heterojunction. (j) In-plane magnetic field dependence of the tunneling conductance. The inset is a local enlargement of the zero-bias region. (k) Field-dependent subgap conductance. Zeeman splitting phenomena can be clearly observed (black dashed lines for eye guidance). Panels (a)-(c) are adapted from Ref. [23]. Panels (e)-(g) are adapted from Ref. [173]. Panels (h)-(k) are adapted from Ref. [174]

One key feature of Andreev bound states is the zero-bias conductance peak of the tunneling junction in the differential conductance measurements, which is similar to that of the Majorana bound state (to be discussed in Sect. 3.4). Therefore, it is important to distinguish the zero-bias conductance peak behavior caused by these two different mechanisms. A theoretical approach to classifying different underlying physical processes for the Andreev reflection and investigating their universal conductance scaling is given as follows [173]. One can see in Fig. 7e-g that the interaction-induced renormalization leads to significantly distinct universal zero-bias conductance behavior, i.e., a distinct characteristic power law in temperature, for different types of Andreev reflections, which show a sharp contrast to that of a Majorana zero mode. These exemplary studies set up a model consisting of semiconducting nanowires and superconducting shells in a superconductor-semiconductor-metal STJ structure (Fig. 7e). Multiple possible processes that might take place in such a structure are considered, and the temperature-dependent zero-bias conductance shows the scaling behavior (power-law relations in Fig. 7g) associated with a dimensionless value $r=R e^{2} /h$, where R is the resistance of the lead resistor and h is Planck constant. Due to their distinct characteristic power laws caused by dissipation, Andreev bound states and Majorana peaks can exhibit different instances of scaling relations. Therefore, the suppression of Andreev bound states and the corresponding zero-bias conductance peak is crucial for the confirmation of Majorana bound states [176].

To effectively induce the Andreev reflection process within STJs, a superconductor-insulator-normal metal structure is always used to modulate the Andreev bound states under an in-plane magnetic field [174]. As depicted in Fig. 7h, i, the embedded defects within the tunnel barrier (semiconducting WSe₂) can serve as quantum dots near the superconductor where Andreev reflection appears. Since the gap value Δ of layered superconductor NbSe₂ is robust under an in-plane magnetic field (Fig. 7j), the degenerate doublet Andreev bound states can undergo Zeeman splitting by an in-plane magnetic field while maintaining the superconducting state of NbSe₂. The subgap near the zero-bias region of a 4-layer NbSe₂ nanoflake varies greatly under the external magnetic field, showing a splitting feature of subgaps (the black dashed lines in Fig. 7k), indicating the magnetic-field-tunable Andreev reflection process therein. This study showed that vdW tunnel junctions could serve as a platform to study the proximity-induced zero-field ground state of quantum dots and its magnetic field tunability.

3.4 Exploration of Majorana bound states in an STJ with layered superconductors

Majorana bound states are promising candidates for non-local and non-Abelian braiding for quantum computation. The Majorana fermion is a unique type of quasiparticle that emerges as its own antiparticle, and it occupies a kind of zero-energy Andreev bound state called the Majorana bound state or Majorana zero mode. Majorana bound states exhibit several captivating characteristics. (1) Majorana bound states can carry both zero spin and zero charge since they can be interpreted as being their own antiparticles (i.e., the creation and annihilation operators associated with a Majorana bound state are identical). (2) Majorana bound states appear precisely at zero energy and are separated from conventional quasiparticle excitations by a distinct energy gap [177]. Due to this property, they are often referred to as Majorana zero modes. (3) Majorana bound states in a 2D host material adhere to neither fermionic nor bosonic quantum exchange statistics [178]. Instead, they exhibit non-Abelian anyonic properties. In simpler terms, exchanging two Majorana bound states leads to a nontrivial rotation of the collective ground state within a degenerate subspace. More importantly, subsequent rotations of such a collective ground state do not necessarily commute. This particular feature makes non-Abelian anyons such as Majorana bound states promising candidates as building blocks for topological quantum computers, where logical operations are executed by exchanging these anyons [179].

In order to experimentally realize Majorana bound states, STJ structures consisting of superconductors and semiconductor nanowires with strong spin-orbital coupling and a large Landé g-factor exhibit great potential in topological quantum computation [167, 180, 182–185]. In the Au/InSb/NbTiN STJ with InSb nanowires (Fig. 8a) [180], the bound states that remained fixed to zero bias were observed even when magnetic fields and gate voltages were changed over considerable ranges. When applying an external field parallel to the one-dimensional (1D) semiconducting nanowire, the four-fold degenerate energy band therein will be split into 2 two-fold degenerate bands; meanwhile, if such a 1D nanowire contacts a superconductor, the proximity effect will induce a superconducting gap Δ in the 1D semiconducting nanowire. Combining the two-fold degenerate energy band and the induced superconducting gap can cause topological superconductivity in the nanowire where Majorana fermions exist. To search for evidence of the presence of Majorana bound states in such a structure, the bias voltage dependence of differential conductance $\mathrm{d} I/ \mathrm{d} V$ under an external magnetic field was studied, as shown in Fig. 8b. Apart from the gap edges marked by green dashed lines, small peaks of differential conductance can be observed near the zero-bias region (highlighted by a green dashed oval). These observations support the hypothesis of Majorana fermions in nanowires coupled to superconductors. Similarly, in a thin InAs/Al hybrid semiconductor-superconductor nanowire device (Fig. 8c) using a four-terminal design [181], zero bias peaks in differential conductance on the order of $2 e^{2} /h$ were observed using tunneling spectroscopy (Fig. 8d), and such a feature can be interpreted as signals of quasi-Majorana zero modes [181, 186–190]. This study suggests that future devices for Majorana bound state investigation should be aimed at longer, thinner, and more uniform InAs/Al wires with better gate and dielectric designs to minimize the level of disorder, which hopefully may lead to better quantization and a more conclusive demonstration of Majorana bound states.

Figure 8

Investigation of Majorana bound states based on STJs. (a) SEM image of the NbTiN/InSb/Au superconductor-semiconductor-normal metal structure. “S” and “N” represent the superconductor and normal metal, respectively. (b) Color mapping of bias voltage- and parallel magnetic field-dependent differential conductance. The green dashed line highlights the zero bias conductance peak, indicating the existence of Majorana fermions. (c) False-color SEM image and lateral structure of the Al/InAs/Au junction. (d) Bias voltage dependence of differential conductance near the quantized conductance $2 e^{2} /h$ under different parallel external magnetic fields. (e) Proximity-induced topological superconductivity in the NbSe₂/WTe₂ heterojunction. Left panel: Temperature-dependent tunnel resistance under different magnetic fields. The inset figure is an optical image of the structure. Right panel: Illustration of the superconductor proximity effect, showing the spatial distribution of superconducting gap Δ in the heterostructure. (f) Zero-bias conductance peak (referred to as “ZBCP” and marked by black arrows in the figure) in the normalized tunneling conductance at different temperatures. Such a feature can be observed without an external magnetic field. Panels (a) and (b) are adapted from Ref. [180]. Panels (c) and (d) are adapted from Ref. [181]. Panels (e) and (f) are adapted from Ref. [147]

The hybrid structure based on a topological insulator and superconductor is also a promising strategy to achieve Majorana bound states [9] for the following two reasons: on one hand, even when there is no applied magnetic field, the interface between the superconductor and the topological insulator is topologically nontrivial. This means that the induced superconductivity on the surface of the topological insulator remains robust even in the presence of disorder, making it feasible to achieve Majorana states at higher temperatures [191]. On the other hand, the surface states of a topological insulator, due to their topological properties, extend throughout the entire boundary of the material. Such extended surface states are essential for realizing Majorana states in topological insulator nanowires [191].

The layered semimetal WTe₂ has recently been found to be a 2D topological insulator when thinned down to a single monolayer, with conducting helical edge channels [194]. In a vdW hybrid structure consisting of NbSe₂ and WTe₂, strong proximity-induced superconductivity in WTe₂ with an ultralong superconducting proximity length was observed [147], and the temperature-dependent tunnel resistance shows two resistance drops at 7.2 K and 6.2 K (Fig. 8e). The former is associated with the superconducting transition of NbSe₂, and the latter originates from the proximity-induced superconducting transition of WTe₂. Note that the observed feature of the zero bias conductance peak (marked by black arrows in Fig. 8f) fades out as the temperature increases, suggesting the existence of Majorana bound states without an external magnetic field. Such observations form a basis in the search for superconducting phases in topological insulators and suggest that topological insulators may lead to intriguing phases of matter, such as Majorana fermions. Experimental results based on STM studies can also indicate the Majorana bound states in the same material system. The scanning tunneling spectroscopy obtained by STM can provide conclusive evidence for proximity-induced superconductivity in the quantum spin Hall edge state in WTe₂ for realizing 1D topological superconductivity and Majorana bound states in these vdW material platforms [195].

4 Josephson junction with Cooper pair tunneling

Unlike the non-zero tunneling resistance observed in STJs with single electron tunneling, a supercurrent with zero resistance can be achieved in Josephson junctions (Fig. 1e). One technical appeal of Josephson junctions is their capability for the precise detection and ultrahigh sensitivity of the magnetic field by determining the critical supercurrent via Cooper pair tunneling therein. Another appeal of Josephson junctions is the non-reciprocal transport of the supercurrent flow in opposite directions and the realization of the superconducting diode effect, which can be achieved by breaking the spatial inversion or time reversal symmetry of the Josephson junction. In the upcoming subsections, we offer an in-depth introduction to the 0-phase or π-phase Josephson junctions, the Andreev reflection and Majorana bound states in Josephson junctions, and the photodetection with Josephson junctions as well as the superconducting diode effect in Josephson junctions.

4.1 Fundamental characteristics of the Josephson junction

To understand the superconducting phase coherence in Josephson junctions, we can represent the superconducting order parameter in the left and right superconductors with $\Psi _{1} = \sqrt{\rho _{1}} \mathrm{e}^{\mathrm{i} \varphi _{1}}$ and $\Psi _{2} = \sqrt{\rho _{2}} \mathrm{e}^{\mathrm{i} \varphi _{2}}$, where $\rho _{1}$ and $\rho _{2}$ are the magnitudes, and $\varphi _{1}$ and $\varphi _{2}$ are the phases of order parameters $\Psi _{1}$ and $\Psi _{2}$ (Fig. 9a, b) [192]. Note that the difference between their phases, $\Delta \varphi = \varphi _{1} - \varphi _{2}$, can be controlled by changing the thickness of the tunnel barrier. The Josephson junction with a phase difference of either $\Delta \varphi = 0$ or π is referred to as a 0-phase or π-phase Josephson junction. Inside the barrier, the supercurrent $I= I_{c } \sin ( \Delta \varphi )$ is caused by the coherence between the two superconductors, where $I_{c }$ is the critical current (Fig. 9c). The Δφ value in $I= I_{c } \sin ( \Delta \varphi )$ can be affected by a magnetic field (d.c. Josephson effect) and electromagnetic wave radiation (a.c. Josephson effect).

Figure 9

Fundamentals of Josephson junctions. (a) Schematic illustration of Cooper pair tunnel in Josephson junctions. (b) Two typical structures for the Josephson junction with layered superconductors. In the lower panel, the Dayhem bridge serves as a weak link between the two superconducting regions, in accordance with Fig. 4. (c) I-V characteristics of the Josephson junction formed by the Nb/InSb/Nb heterostructure. Note that a hysteresis loop can be confirmed. (d) Asymmetric d.c. Josephson effect in inversion symmetry breaking topological Josephson junctions. The upper and lower panels are theoretical calculation results for type-I and type-II Weyl semimetals. (e) Fractional a.c. Josephson effect in the Nb/InSb/Nb Josephson junction under 3 GHz electromagnetic radiation. The Shapiro steps with a height $\Delta V=h\nu /2e$ (ν is the frequency of electromagnetic radiation) can be observed for $B<2 T$. Panel (a) is adapted from Ref. [192]. Panels (c) and (e) are adapted from Ref. [16]. Panel (d) is adapted from Ref. [193]

In the d.c. Josephson effect, when an external magnetic field is applied perpendicular to the direction of Cooper pair tunneling, the $I_{c }$ value of the Josephson junction depends on the magnitude of the magnetic field B:

$$\begin{aligned}& I_{c } ( B ) = I_{c } ( 0 ) \biggl\vert \frac{\sin ( \frac{\pi \Phi _{J }}{\Phi _{0}} )}{ \frac{\pi \Phi _{J }}{\Phi _{0}}} \biggr\vert \end{aligned}$$

(1)

where $I_{c } ( 0 )$ is the supercurrent of the Josephson junction under zero magnetic field, $\Phi _{J } \propto B$ is the value of magnetic flux penetrating through the tunnel barrier, and $\Phi _{0} =h/ 2e$ is the quantized magnetic flux. Equation (1) can provide a curve similar to the Fraunhofer diffraction results [196–199]. Interestingly, an asymmetric Fraunhofer pattern (the magnitude of $I_{c }$ across the junction does not equal the critical current flowing in the opposite direction at fixed magnetic field B) is expected to exist when a topological material with inversion symmetry breaking serves as the tunnel barrier [193, 200, 201], as shown in Fig. 9d. Such a phenomenon is crucial for the realization of superconducting diodes (discuss in Sect. 4.5). For the a.c. Josephson effect typically achieved with the incidence of electromagnetic waves, one can clearly see the quantized voltage steps (Shapiro steps) in the I-V characteristics [202–204]. Taking the Nb/InSb/Nb Josephson junction as an example, when the junction is irradiated with a radio frequency $f_{0}$ in the absence of an external magnetic field, the observed Shapiro steps with a height $\Delta V = hf_{0}/2e$ are expected for conventional Josephson junctions, where the supercurrent is carried by charge-2e Cooper pairs. At high magnetic fields, the height of the first Shapiro step is doubled to $hf_{0}/e$ (referred to as the fractional a.c. Josephson effect), suggesting that the supercurrent is carried by charge-e quasiparticles (Fig. 9e), serving as a unique signature of the Majorana fermions [16].

4.2 0-phase and π-phase Josephson junctions

In Josephson junctions, the interaction between ferromagnetism and superconductivity has led to the emergence of an unconventional π-phase ground-state junction in superconductor-ferromagnet-superconductor junctions. Such π-phase junctions can occur for ferromagnets with specific thicknesses due to the quantum intermixing of wavefunctions between spin-singlet Cooper pairs in the superconductor and spin-polarized electrons in the ferromagnet [205, 206]. At the interface between the ferromagnet and superconductor, when a Cooper pair penetrates into the ferromagnetic material, it possesses a measurable center-of-mass momentum. This particular momentum leads to fluctuations in the real part of the superconducting order parameter, as schematically illustrated in Fig. 10a. Since these fluctuations can be controlled by varying the ferromagnetic layer thickness (Fig. 10b) [161, 207], the wavefunctions of the Cooper pairs in the two superconductors flanking the ferromagnet can either have a phase difference of 0, resulting in “0-phase junctions” with positive Josephson coupling (Fig. 10c), or have a phase difference of π, forming “π-phase junctions” with negative Josephson coupling (Fig. 10d).

Figure 10

Investigation of 0-phase and π-phase Josephson junctions. (a) Schematic illustration of 0-phase (left) and π-phase (right) Josephson junctions. (b) Oscillatory real part of the superconducting order parameter ($\mathrm{Re}\{\Psi\}$, green curve) penetrating ferromagnets (represented by FM in the figure) leads to the 0-phase state and π-phase state. (c) Symmetric order parameter in the 0-phase junctions. (d) Antisymmetric order parameter in the π-phase junctions. (e) Transmission electron micrographs measured on the typical Nb/NiFe (12 nm)/Nb sample. (f) NiFe thickness dependence of the Gilbert damping difference (Δα) between the Nb/NiFe/Nb π-phase and 0-phase junctions at 4 K. Inset: Gilbert damping of 0-phase and π-phase junctions. (g) Illustration of spin pumping via Andreev bound states and enhanced Gilbert damping for Nb/NiFe/Nb π-phase junctions. (h) Illustration of spin pumping via the Andreev bound state and enhanced Gilbert damping for Nb/NiFe/Nb 0-phase junctions. (i) Schematic cross section of the Nb/Cu_1 − xNi_x/Nb junction. (j) Critical current $I_{c }$ as a function of temperature T for two junctions with Cu_0.48Ni_0.52 and $d_{F} = 22$ nm. Inset: $I_{c }$ versus magnetic field H for the temperatures around the crossover to the π-phase state as indicated on curve b): (1) 4.19 K, (2) 3.45 K, (3) 2.61 K. Panels (b)-(h) are from Ref. [93]; Panels (i) and (j) are from Ref. [208]

To investigate the interplay between ferromagnetism and superconductivity within the Josephson junction, giant oscillatory Gilbert damping in superconducting Nb/NiFe/Nb junctions has been demonstrated [93], and further suggests unconventional spin pumping and relaxation via zero-energy Andreev bound states that exist in 0-phase junctions and π-phase junctions (Fig. 10e). Based on the definition of the Gilbert damping constant (the rate at which a magnetic moment in a ferromagnetic material relaxes or dampens in response to an external magnetic field), the oscillatory Gilbert damping was found to be dependent on the thickness of the NiFe layer and was not influenced by changes in temperature. This substantial oscillation in Gilbert damping can be explained by the mechanism of spin pumping via Bogoliubov quasiparticle-mediated spin currents in superconductors. As the temperature decreases, there is a consistent and rapid decline in the population of Bogoliubov quasiparticles due to the expanding superconducting gap. Consequently, this reduction leads to lower Gilbert damping and results in smaller oscillations when compared to conditions observed in normal states. The most reasonable mechanism is spin pumping via the Andreev bound state.

For π-phase junctions, the Andreev bound states locate near zero energy levels within the superconducting gap. The Andreev bound states could penetrate into the superconducting Nb films with scale of superconducting coherent length, which is evanescent to dissipate the spin angular momentum [209, 210]. When spin angular momentum is transferred via these zero-energy Andreev bound states, it can provide an additional Gilbert damping constant $\alpha _{\mathrm{SP}}$ to the Gilbert damping from the ferromagnet layer itself $\alpha _{\mathrm{FM}}$ (Fig. 10g). For 0-phase junctions, the Andreev bound states are distributed near the edge of the superconducting gap (Fig. 10h). At lower temperatures, fewer Andreev bound states are available, so the efficiency of spin pumping is reduced. Therefore, as the thickness of the NiFe layer increases, there is an oscillatory pattern in the efficiency of spin pumping via Andreev bound states at the interface (Fig. 10f). Such findings could be important for further exploring emergent phenomena with ferromagnet π-phase junctions in quantum computing, such as half-quantum flux qubits.

Instead of modulating the thickness of the ferromagnetic layer to induce the phase shift from 0 to π mentioned above, the 0-to-π phase shift in STJs can also be achieved by controlling the temperature [208, 211]. For example, in Nb/Cu_xNi_1 − x/Nb junctions (Fig. 10i), a phase change from 0 to π can be achieved when cooling down the temperature. The critical tunneling supercurrent $I_{c }$ monotonically increases with decreasing temperature, but a maximum in $I_{c }$ is achieved with a strong decrease down to zero. Such a sharp cusp in $I_{c }$ can be explained only by assuming that the junction changes from a 0-phase state at high temperatures to a π-phase state at low temperatures. In this research, only the absolute value of the current was measured, and the 0-to-π phase shift was revealed as a sharp cusp in the $I_{c }$-T relation (Fig. 10j). This research offers an easy way to realize π-phase junctions and may be of significance to superconducting quantum computing since several schemes for the necessary qubits (quantum two-level systems) rely on phase shifts of π in a superconducting network.

4.3 Andreev reflection and Majorana bound states in Josephson junctions

The tunneling supercurrents in Josephson junctions are in principle enabled by Andreev scattering, making Josephson junctions an ideal platform for studying Andreev bound states (Fig. 11a) and Majorana bound states. In Nb/Cd₃As₂/Nb Josephson junctions (Fig. 11b), proximitized superconductivity can be induced in sandwiched Cd₃As₂. The observed critical current oscillations can be attributed to the existence of surface Fermi arcs in the band structure of Dirac semimetal Cd₃As₂ [212]. The proximity-induced supercurrent from bulk states of Cd₃As₂ is suppressed under an in-plane magnetic field of ∼0.1 T, while the supercurrent from the topological surface states of Cd₃As₂ survives up to 0.5 T. Contrary to the minimum normal-state conductance of Cd₃As₂, the supercurrent carried by Fermi-arc shows a maximum critical value near the Dirac point, which is consistent with the fact that the Fermi arcs have a maximum density of states at the Dirac point [212]. Note that the critical current exhibits periodic oscillations with a parallel magnetic field, which is well understood by considering the in-plane orbital effect from the surface states of Cd₃As₂. Such results suggest that the Dirac semimetal combined with superconductivity should be promising for topological quantum devices. Importantly, a series of dips at $V_{n} = 2\Delta /ne$ in the differential resistance-$V_{gs}$ curve can be attributed to multiple Andreev reflections at the interface (Fig. 11b). This work demonstrates flexible gate and magnetic field manipulations of Fermi-arc superconductivity and provides a good platform for the investigation of superconductivity of chiral polarized states [212].

Figure 11

Andreev reflection in in-plane Josephson junctions. (a) Schematic illustration of Andreev bound state in a typical Josephson junction device. (b) The differential resistance dV/dI versus source-drain voltage $V_{{dc}}$ across the junction, showing the signatures of multiple Andreev reflections (highlighted by dashed lines). Inset: Schematic illustration of the Nb/Cd₃As₂/Nb Josephson junction. (c) Cross-sectional schematic view (upper panel) and false-colored SEM image (lower panel) of the Al/InAs/Al Josephson junction. (d) Differential resistance dV/d$I_{b}$ as a function of $V_{{bg}}$ and $I_{b}$. (e) Differential conductance d$I_{{sd}}$/d$V_{{sd}}$ as a function of source-drain voltage $V_{{sd}}$ measured at $V_{{bg}}= 0$ V, $B = 0$ T at the base temperature T∼ 20 mK. The vertical gray dashed lines mark the positions of multiple Andreev reflection peaks, and the integer numbers n represent the orders of the multiple Andreev reflections. Panel (b) is adapted from Ref. [212]. Panels (c)-(e) are from Ref. [22]

Multiple Andreev reflections (introduced in Sect. 3.3) can serve as evidence of the high interface transparency of a Josephson junction. For example, in planar Josephson junctions made from InAs nanosheets (Fig. 11c), subharmonic peaks at $n = \pm1$, ±2, and ±4 in the differential conductance were achieved, confirming the multiple Andreev reflections at the interface [22]. The central dark region in the differential resistance-bias current ($I_{b}$) relations (Fig. 11d) indicates that the junction is in the superconducting state, while the dark fringes caused by multiple Andreev reflections are easily visible on the outer edges of this central area. Distinct peaks with positions aligning closely with $eV_{sd}(n) = 2\Delta /n$ can be clearly observed in the differential conductance-source-drain voltage relation (Fig. 11e) diagram, where n takes values such as ±1, ±2, and ±4. These peaks are also a consequence of multiple Andreev reflections. The observation of multiple Andreev reflections extending up to $n = \pm 4$ suggests both excellent quality and high transparency at the interface of the Josephson junction with the InAs nanosheet. The excellent proximity-induced superconducting properties observed in these Josephson junctions with InAs nanosheet show that InAs nanosheet-based hybrid devices are desired systems for the realization and manipulation of topological superconducting quantum states.

A Josephson junction geometry with a topological insulator film as the tunnel barrier (Fig. 12a) was used to exploit the unique helical and extended nature of the topological insulator surface state [213]. When the magnetic flux through the junction is close to an integer number of flux quanta, Majorana states, present on both surfaces of the film, should give rise to a narrow peak-dip structure in the current-phase relation by hybridizing at the edge of the junction. In this theoretical model, superconductivity exists on both surfaces of the topological insulator film, which can be achieved by both surfaces contacting the superconductor, or by one surface contacting the superconductor and transmitting superconductivity to the other surface through bulk states or side surfaces [215]. According to the calculated current-phase (I-θ) relations under different magnetic fluxes through the junction, the presence of Majorana zero modes starts to remarkably affect the system when the magnetic flux is approximately within 5% of a single flux quantum $\Phi _{0}$. Initially, this contribution is seen as a slight deviation from the background sinusoidal curve but gradually becomes more pronounced and becomes a bump within the 1% range of $\Phi _{0}$ [213]. The relation between the critical current $I_{c }$ and magnetic flux (Fig. 12b) closely resembles the Fraunhofer pattern described by Eq. (1) in Sect. 4.1. However, it deviates from the expected behavior at integer flux values because of the influence of Majorana fermion contributions to the Josephson current, causing a non-zero $I_{c }$ value. Another theoretical calculation on Josephson junctions with topological insulators revealed a similar I-θ relation (Fig. 12c) [140]. The features shown in these studies point out a characteristic signature of Majorana states based entirely on supercurrents of the Josephson junction, adding a significant piece in the theoretical foundation for experimentally identifying Majorana bound states in Josephson junctions.

Figure 12

Majorana bound states in the Josephson junction with layered superconductors. (a) Depiction of a Josephson junction with topological insulator. The upper surface of the topological insulator (gray layer) is in contact with the superconductors (blue layers), and the superconductivity is transmitted to the bottom surface through the bulk states. As the global-phase offset between the left and right superconductors ($\theta _{0} = \theta _{R } - \theta _{L }$) is adjusted, Majorana modes (shown as red circles) bound to Josephson vortices are created at one end of the junction, move along the junction, and fuse on the opposite side of the junction. (b) Current-phase relationship of the S-TI-S junctions. Each curve is labelled by the value of ${\Phi _{{B}}} / {\Phi _{0}}$. The Majorana contribution to the Josephson current appears as a shoulder in the curves and eventually dominates close to $\Phi _{{B}} = \Phi _{0}$. (c) Theoretical current-phase relation. The composite current-phase relation is shown in black, with the Majorana contribution shown in red. (d) Critical current dependence on the magnetic field of Nb/Bi₂Te₃/Nb Josephson junctions. Inset: SEM image of Nb/Bi₂Te₃/Nb Josephson junctions. (e) Differential resistance, dV/dI, plotted on a colour scale as a function of the bias current and the 10 GHz microwave excitation power $P_{\mathrm{RF}}$. The numbers correspond to the nth order of the Shapiro step. (f) Temperature dependence of $I_{c }$. The black solid line and dark blue dashed line are fitting curves based on the ballistic transport of Eilenberger theory and diffusive transport of Usadel theory, respectively. Panels (a) and (b) are from Ref. [213]. Panel (c) is adapted from Ref. [140]. Panels (d)-(f) are from Ref. [214]

An essential step in the experimental hunt for the Majorana fermion is the unequivocal observation of supercurrent in a topological phase. In Nb/Bi₂Te₃/Nb junctions, as expected from theoretical predictions, both clear Shapiro steps in the I-V curve under microwave irradiation and a Fraunhofer-type critical current can be observed (Fig. 12d, e) [214]. Although the electronic transport therein is a ballistic process described by Eilenberger theory rather than a diffusive process (Fig. 12f), the mean free path of the charge carrier is only approximately 22 nm, which is too small to induce such a ballistic transport phenomenon. Therefore, the ballistic supercurrent in the Josephson junction is supposed to originate from the topologically protected surface state associated with Majorana bound states. The realization of supercurrents through topological surface states is an important step towards the detection of Majorana fermions.

Similar to vdW Josephson junctions, planar Josephson junctions offer an alternative approach for confining Majorana channels therein. The phase difference Δφ between the superconducting leads represents an additional tuning knob that is predicted to drive the system into the topological phase at lower magnetic fields than for a system without phase bias [216]. The phase and magnetic-field dependence of the zero-energy states in the Al/InAs/Al Josephson junction has been shown to be consistent with a model of Majorana zero modes in finite-size Josephson junctions [216]. A robust zero-bias peak exhibiting a strong dependence on Δφ was measured by tunneling spectroscopy using a laterally coupled quantum point contact. Biasing the junction to $\Delta \varphi \approx \pi$ reduces the critical field at which the zero-bias peak appears with respect to $\Delta \varphi = 0$. These experiments showing excellent agreement with theoretical investigations offer support for the potential creation of Josephson junctions that can accommodate Majorana fermions and underscore the importance of exploring Majorana bound states in topological surface states.

4.4 Photon radiation and photodetection based on Josephson junctions

Since the frequency emitted from Josephson junctions follows the mechanism of the a.c. Josephson effect, the subterahertz (sub-THz) radiation can be achieved based on Josephson junctions, showing great feasibility for non-destructive inspections, medical diagnostics, and high-speed communications [217]. In particular, the integer higher harmonics observed in all spectra imply that the a.c. Josephson effect plays the dominant role in the novel dual-source radiation mechanism. For conventional superconductors, due to the low $T_{c }$ and small Δ value, the emitted photons can only have a frequency of several hundreds of GHz since 1 meV of the Δ value corresponds to approximately 483.6 GHz of a photon. However, the high-$T_{c }$ superconductor Bi₂Sr₂CaCu₂O_8 + δ has a gap value of ∼10 times that of conventional superconductors, thus being able to emit photons with a frequency of several THz [217]. The photon emission of the short rectangular-, square-, and disk-shaped mesas of intrinsic Josephson junctions based on Bi₂Sr₂CaCu₂O_8 + δ (Fig. 13a) was fabricated to investigate the radiation spectra (Fig. 13b), where the second harmonic radiation frequency can be observed for each emitter, indicating the contribution from the a.c. Josephson effect. Based on THz (or sub-THz) radiation measurements from mesas of various geometrical shapes, a dual-source radiation mechanism is proposed: the uniform part of the a.c. Josephson current is the primary radiation source, and a single electromagnetic cavity mode is the secondary source. These experiments have effectively demonstrated sub-THz radiation using Josephson junctions, paving the way for innovative devices in quantum microwave generation.

Figure 13

Emitter and photodetector based on the Josephson junction with layered superconductors. (a) Schematic illustration of the intrinsic Josephson junction with Bi₂Sr₂CaCu₂O_8 + δ (marked as Bi-2212 in the figure). Three shapes of the intrinsic Josephson junction (marked as “IJJ” in the figure) were investigated, as shown in the SEM photos in the lower panels. (b) Emission spectrum for the above-mentioned 3 structures at 25 K, detected by a Fourier transform infrared spectrometer. The double-frequency signal for each emitter can be clearly confirmed. Inset: the $1/(2a)$ dependence of the fundamental frequency, where a is the radius of mesa shown in (a). (c) Schematic illustration of photodetection mechanism based on Josephson junction device. (d) Schematic of the graphene-based Josephson junction. (e) Switching events in the dark and 100 pW of 1550-nm light at $I_{b} = 10.90$ mA. (f) The observed switching events under illumination (0 to 200 pW) follow Poisson statistics. (Inset) The statistical variance of counts $\sigma _{m}^{2}$, the photon shot noise, equals its mean mm without fitting (solid line). (g) The switching probability as a function of the average absorbed photon number $\langle N_{\mathrm{photon}} \rangle$ in a 20-μs time window. The switching probability is linearly proportional to $\langle N_{\mathrm{photon}} \rangle$ with an offset owing to the dark count. The solid line shows the fitting weighted with the inverse of standard deviation (error bars). Panels (a) and (b) are adapted from Ref. [217]. Panels (d)-(g) are adapted from Ref. [42]

Since the performance of Josephson junction is sensitive to photon-induced Cooper pair de-pairing, as observed in qubit relaxation [218, 219], high-sensitivity photodetection can be feasible based on such mechanism [41, 220–224], as illustrated in Fig. 13c. Single near-infrared photon detection has been realized by coupling photons to the localized surface plasmons of a NbN/graphene/NbN Josephson junction (Fig. 13d). Using the photon-induced switching statistics of the current-biased device, the quasiparticles generated by the absorbed photon were revealed to be critical in the detection mechanism [42], where a single near-infrared photon absorbed into the superconductor breaks Cooper pairs and generates decoupled electrons, which then become a noise source to switch the current-biased Josephson junction from the superconducting state to the normal state with a non-zero voltage across it [225]. Figure 13e shows the switching events registered when the Josephson junction was exposed to near-infrared photons and when it was not. After collecting 10⁴-second time traces at various power levels, the data were used to create histograms to analyze the occurrence of switching events using 1-second bins (Fig. 13f). The experimental data closely follow a Poisson distribution (solid lines in Fig. 13f), which is expected for uncorrelated switching events. The inset in Fig. 13f also illustrates a key property of Poisson statistics: the variance in the count is equal to the mean. It was also observed that the probability of a switch occurring within a 20-millisecond time window, accounting for an offset caused by dark counts, is directly proportional to the average number of absorbed photons $\langle N_{\mathrm{photon}} \rangle$ (Fig. 13g). This finding demonstrates that the designed STJ effectively detects single photons from a low-intensity coherent source [42].

4.5 Superconducting diode based on Josephson junctions

A superconducting diode refers to a device that allows for the flow of supercurrent in one direction but obstructs the flow of the supercurrent along the opposite direction (Fig. 14a). To achieve such a non-reciprocal transport process, a superconducting diode is usually associated with a broken space or time reversal symmetry. Superconducting diodes could be used in superconducting quantum electronics, similar to the situation in which regular diodes are currently utilized in electrical circuits [25, 126]. Josephson junction-based superconducting diodes enable directional charge transport without energy loss at low temperatures, leading to an ultrahigh sensitivity detection circuit and a modulator with ultralow power consumption [97, 227–231].

Figure 14

Superconducting diodes based on Josephson junctions. (a) Schematic illustration of the general mechanism of superconducting diode effect in Josephson junction devices. (b) Schematic illustration of a superconducting diode achieved by Josephson junctions. A non-zero field is typically needed. (c) Schematic of the Nb/NiTe₂/Nb Josephson junction. (d) I–V characteristics of the Josephson junction with an in-plane magnetic field $B_{{y}} = 20$ mT showing a large non-reciprocal critical current, namely, $I_{c-}$ and $I_{c+}$. (e) Rectification effect observed using currents between $| I_{c -} |$ and $I_{c+}$. (f) The I–V characteristics for the Nb/CuNi/Nb Josephson junction under three different values of the out-of-plane magnetic field. A profound nonreciprocity between $I_{c+}$ and $| I_{c -} |$ can be seen. Inset: SEM image of the Nb/CuNi/Nb Josephson junction. (g) Illustration of different symmetry requirements of the superconducting diode. (h) The I–V characteristics for the out-of-plane NbSe₂/Nb₃Br₈/NbSe₂ Josephson junction (structure and optical image shown in Fig. 3e). Apparent discrepancies between $I_{c +}$ and $\vert I_{c -} \vert$, as well as $I_{{r}+}$ and $\vert I_{{r}-} \vert$, can be observed. Panel (b) is adapted from Ref. [25]. Panels (c)-(e) are adapted from Ref. [31]. Panel (f) is adapted from Ref. [130]. Panel (g) is adapted from Ref. [226]. Panel (h) is adapted from Ref. [24]

Superconducting diode with broken time reversal symmetry

An external magnetic field can break the time reversal symmetry of a Josephson junction (Fig. 14b) to generate non-reciprocal electronic transport behavior [25, 232]. The underlying mechanism is that, in non-centrosymmetric superconductors, Cooper pairs tend to acquire a center-of-mass momentum when subjected to an external magnetic field. Such finite-momentum pairing can result in an asymmetric critical current behavior, where a supercurrent can propagate along one direction but not in the opposite direction. For example, in Josephson junctions with a type-II Dirac semimetal NiTe₂ (Fig. 14c), a giant Josephson diode effect has been demonstrated and explained by the finite-momentum Cooper pairing mechanism that largely originates from the Zeeman shift of spin-helical topological surface states. The d.c. I-V curve obtained under a magnetic field perpendicular to the current direction of 20 mT (Fig. 14d) presents four different critical current values: during the negative-to-positive current sweep, the device exhibited two critical current values, $I_{\mathrm{r}-}$ and $I_{c +}$, while during a positive-to-negative current sweep, the device exhibited two other critical current values, $I_{\mathrm{r} +}$ and $I_{c -}$. These distinct $I_{c +}$ and $| I_{c -} |$ values mean that when the absolute magnitude of the applied current falls between $I_{c +}$ and $| I_{c -} |$, the STJ behaves as a superconductor for current flowing in one direction while acting as a normal dissipative metal for current in the opposite direction, showing a rectification effect for currents larger than $I_{c +}$ but smaller than $| I_{c -} |$ (Fig. 14e). The observed giant magnitude of the asymmetry in critical current and the clear exposition of its underlying mechanism pave the way to build novel superconducting computing devices using the Josephson diode effect.

Superconducting diode with broken space/time symmetry

Since the Abrikosov vortices can break the space/time symmetry, an out-of-plane magnetic field applied on a type-II superconductor can also lead to the superconducting diode effect [130, 233]. In Nb/CuNi/Nb Josephson junctions (Fig. 14f), the trapped Abrikosov vortices have been demonstrated to break the space/time symmetry and result in the non-reciprocal supercurrent [130]. The nonreciprocity of critical current $\vert \frac{I_{c +}}{I_{c -}} \vert$ in such diodes can reach up to a factor of 10 and can be easily switched on or off by trapping or removing the Abrikosov vortices. Such a switchable feature enables a memory functionality with three distinct states even without a magnetic field. Note that the rectification efficiency exceeds 70% at zero field, which is sufficient for realization of more complex logical Boolean devices needed for a digital superconducting computer. Such rectification behavior facilitates a memory functionality in superconducting diodes and can potentially be used for in-memory superconducting computers [234, 235].

Superconducting diode with broken space inversion symmetry

Unlike a superconducting diode with broken time reversal symmetry by an external magnetic field, a superconducting diode with broken space inversion symmetry can realize non-reciprocal supercurrent transport without the magnetic field. Theoretical analysis predicts that if the Josephson junction has a tunnel barrier with broken space inversion symmetry, the non-reciprocal transport behavior can be observed (Fig. 14g) [226]. Experimentally, the superconducting diode effect has been demonstrated in an inversion symmetry breaking vdW heterostructure of NbSe₂/Nb₃Br₈/NbSe₂ (Fig. 3e), which, even without a magnetic field, can be superconducting with a positive current while being resistive with a negative current [24]. The intentionally chosen 3-layer Nb₃Br₈ broke the inversion symmetry (confirmed by the antisymmetric second harmonic generation results), and the corresponding I-V characteristics indeed have different $I_{c +}$ and $\vert I_{c -} \vert$ values, as presented in Fig. 14h. Importantly, stable half-wave rectification of a square-wave excitation was achieved with a very low switching current density, high rectification ratio and high robustness. This non-reciprocal behavior opens the door to discover new mechanisms and physical phenomena through the integration of quantum materials with Josephson junctions and provides new avenues for superconducting quantum devices. Such a result is an excellent paradigm of “symmetry engineering” based on layered materials [236–239] and has sparked a large number of subsequent works [29, 240–242].

5 Summary and outlook

In this review, we comprehensively surveyed forefront research fields in layered STJs by covering Andreev scattering, Majorana zero modes, and the superconducting diode effect, as well as 0-phase and π-phase Josephson junctions. The typical layered superconductors for the formation of STJ structure include 2D crystalline van der Waals superconductors (2H-NbSe₂, Bi₂Sr₂CaCu₂O_8+δ, etc.) and thin layer of evaporated/sputtered superconductors (Al, Nb, NbTi, NbTiN, etc.). Since the superconducting gap and $T_{c }$ values are different in these superconductors, the STJ devices have different working temperatures and different characteristic parameters. Also, the crystallinity of a superconductor also affects its properties. For the investigation of superconducting proximity effect, although sputtering a thick superconducting layer like Nb on the targeted material can form the STJ structure, the polycrystalline nature of sputtered material significantly weakens the proximity effect across the interface. Replacing the sputtered superconductor with crystalline superconductor like 2H-NbSe₂ to form the STJ structure can effectively enhance the proximity effect therein. On the other hand, the typical tunnel barriers applied in STJ structures include quasi-1D semiconductors with large spin-orbital coupling effects and 2D van der Waals semiconductors, as well as topological semimetal and other materials, depending on the functionalities of the devices. To better understand the characteristics of each kind of tunnel barrier, we summarize the typical tunnel barrier materials used in STJs and their functionalities in Table 1. One can see that some tunnel barrier materials can have multiple capabilities due to their unique band structures therein.

Table 1 Summary of typical tunnel barrier materials and their functionalities in STJ devices

Currently, STJs are not only recognized as a device geometry to fundamentally understand Cooper pair coherence, topological states and superconducting gaps, but also serve as an important component for non-reciprocal supercurrent transport, half-quantum flux qubits and photon detection. However, there is still plenty of room to develop STJs towards understanding emerging pairing states, phase coherence between superconducting order parameters, experimental realization of Majorana bound states, and quantum computation. The challenges and limitations of current STJ devices can be summarized as follows: For experimental setups, the fabrication of STJs with clean interface and high performance remains challenging. Besides, the precise measurement techniques (for instance, zero-bias conductance peak) to capture the exotic quantum states in the STJs, as well as accurate controlling of single quantum flux and the braiding of vortices in STJ devices remain elusive. Also, the growth of these crystalline heterostructures with clean and sharp interfaces based on molecular beam epitaxy and other approaches remains challenging and requires further exploration. However, it should also be addressed that the breakthroughs in fabrication and measurement of STJ devices have extremely large scalability. Since the STJ is a family member of tunnel junction devices, the techniques used in STJ devices can be applied for fabrication and measurement of magnetic tunnel junctions, ferroelectric tunnel junction and multiferroic tunnel junctions. Also, the highly precise techniques to measure the tunneling conductance is also helpful for the studies of other tunnel junction devices.

Although many challenges still exist in the mechanisms and applications of STJs, there are plenty of astonishing discoveries and rapid progress of this field. Potential breakthroughs of STJs can be expected in the following aspects. On the field of topological quantum computation, potential breakthroughs include: (a) In the aspect of theoretical exploration, the understanding and suppression of Andreev bound states in the investigated system, as well as finding other material systems for topological quantum computation like parafermions, fractional quantum Hall states, and possible Majorana quasiparticles in Kitaev model would be crucial for quantum computation. Besides, the methodology of quantum error correction and fault-tolerant quantum computation require to be established; (b) In the aspect of device fabrication, apart from the Majorana zero modes in superconductor/semiconducting nanowire heterostructures, controllable growth of high-quality artificial heterojunctions of topological superconductors and planar Josephson junctions would be beneficial for high-quality quantum computation devices; (c) In the aspect of experimental measurements, precise measurements of quantized zero-bias differential conductance, generating and controlling the quantized vortex, as well as determining and tuning the number of electrons in Majorana island of STJ structure are crucial for the application of topological quantum computation; On the field of candidate materials of STJ devices, since STJs with high-$T_{c }$ and highly crystalline superconductors can have higher working temperatures and more fascinating physical phenomena, the searching of high-$T_{c }$ superconductors can provide potential breakthroughs in STJ devices. Besides, the pairing mechanisms of novel unconventional superconductors (for instance, Sr₂RuO₄, nickelates, and Na_xCoO$_{2}\cdot y$H₂O) remains unclear. Potential breakthroughs on the pairing mechanism and pairing symmetry may be achieved by forming STJ structures with these superconductors. We briefly summarize our perspectives on the key challenges and a few potential research directions in the research of STJs with layered superconductors as follows:

• (1) Experimental realization of Majorana bound states based on STJs. One captivating avenue where STJs promise to make significant contributions is the exploration of Majorana fermions based on topological superconductors [246]. The ultimate proof of topologically protected Majorana fermions can be offered by braiding experiments, which can reveal non-Abelian statistics [180]. Braiding can be implemented by using a variety of knobs, such as gate voltages, charging energies or supercurrents [8, 167, 247]. The most direct implementation relies on actually braiding Majorana states in real space [8, 248]. Alternatively, braiding can be performed in parameter space, using couplings to auxiliary Majorana, either by controlling the couplings directly or by using Coulomb blockade to mediate interactions [8, 249–251]. The non-abelian nature of Majorana bound states, combined with Coulomb charging effects, provides a starting point for numerous intriguing constructions including error-correcting networks in quantum computation and for realization of strongly interacting models such as the Kitaev spin models, Sachdev–Ye–Kitaev models or for realizations of supersymmetric models [8, 252–259].

• (2) Quantum computation utilizing STJs. Another field in which STJs show promising applications is quantum computation. The switching behavior of a current-biased Josephson junction around its critical current serves as a widely employed threshold discriminator of two qubit states [260], and the existence of Majorana bound states offers an avenue for the realization of topologically protected quantum gates [248, 261]. These gates enable robust quantum operations that are immune to perturbations, offering a potential solution to the decoherence problem. Experiments focused on the refinement and optimization of gate control are a highly compelling pursuit. This endeavor holds the potential to drive significant advancements in the field of quantum computing [262]. Nonetheless, the integration of STJs into quantum computation presents many challenges. Since the braiding in Majorana-based systems alone does not provide a universal gate set, a proposed solution for this problem involves the introduction of an additional single-qubit phase gate paired with the readout of states encoded by quartets of Majorana modes [263]. Assessing the fidelity of such a phase gate would constitute a worthwhile experimental endeavor.