What can Andreev bound states tell us about superconductors?

Abstract

Zero-energy Andreev bound states, which manifest themselves in the tunnelling spectra as zero-bias conductance peaks (ZBCPs), are abundant at interfaces between superconductors and other materials and on the nodal surface of high-temperature superconductors. In this review, we focus on the information such excitations can provide on the properties of superconductor systems. First, a general introduction to the physics of Andreev bound states in superconductor/normal metal interfaces is given with a particular emphasis on why they appear at zero energy in d-wave superconductors. Then, specific spectroscopic tunnelling studies of thin films, bilayers and junctions are described, focusing on the corresponding ZBCP features. Scanning tunnelling spectroscopy (STS) studies show that the ZBCPs on the c-axis YBa₂Cu₃O_7−δ (YBCO) films are correlated with the surface morphology and appear only in proximity to (110) facets. STS on c-axis La_1.88Sr_0.12CuO₄ (LSCO) films exhibiting the 1/8 anomaly shows spatially modulated peaks near zero bias associated with the anti-phase ordering of the d-wave order parameter predicted at this doping level. ZBCPs were also found in micrometre-size edge junctions of YBCO/SrRuO₃/YBCO, where SrRuO₃ is ferromagnetic. Here, the results are consistent with a crossed Andreev reflection effect (CARE) at the narrow domain walls of the SrRuO₃. ZBCPs measured in STS studies of manganite/cuprate bilayers could not be attributed to CARE because the manganite's domain wall is much larger than the coherence length in YBCO, and instead are attributed to proximity-induced triplet-pairing superconductivity with non-conventional symmetry. And finally, ZBCPs found in junctions of non-intentionally doped topological insulator films of Bi₂Se₃ and the s-wave superconductor NbN are attributed to proximity-induced p_x+ ip_y triplet order parameter in the topological material.

This article is part of the theme issue ’Andreev bound states’.

1. Introduction

It is well known that zero-energy Andreev bound states (ABSs) reside on the nodal surface of d-wave superconductors, e.g. on the (110) surface of the cuprate high-temperature superconductor (HTSC) YBa₂Cu₃O_7−δ (YBCO). The corresponding zero-bias conductance peak (ZBCP) observed in tunnelling spectra measured on such surfaces is, in fact, one of the hallmarks of d-wave superconductivity (we note, in passing, that ‘d’ stands for ). Such spectral features, first discussed by Hu [1], have been observed by many groups using various methods, such as point contact spectroscopy, planar junctions and edge junction geometries. The tunnelling spectra could be well fitted to the theory of tunnelling into a d-wave superconductor developed by Tanaka & Kashiwaya [2] and Kashiwaya et al. [3], who extended the Blonder–Tinkham–Klapwijk (BTK) model [4] developed for an s-wave superconductor. In brief, the origin of the zero-energy ABS and the corresponding ZBCP can be understood within a simple model, assuming a very narrow normal region formed at the ‘pair-breaking’ nodal surface of the d-wave superconductor. Within this region, multiple alternating Andreev and normal reflection processes can take place, as demonstrated in figure 1, where in the Andreev reflection, in addition to the ‘conventional’ π phase shift (for zero-energy quasi-particles), there is another phase shift acquired due to the sign change in the order parameter experienced by the incoming and outgoing quasi-particle. With that, a zero-energy ABS can be formed, irrespective of the thickness of the normal surface region [1,5]. As we shall show in this paper, the ABS can tell us more about the properties of superconductors and their interfaces with other materials, beyond the mere existence of d-wave order-parameter symmetry in the cuprates. Before this, however, we will elaborate somewhat more on their formation.

Figure 1.

Illustration of a closed quasi-particle trajectory comprising two Andreev and two normal reflections that gives rise to ABS. In this specific case of a d-wave superconductor where the surface normal is along the nodal direction (α = π/4), the ABS has zero energy. (Online version in colour.)

The concept of an ABS was first discussed for the case of s-wave superconductors long before the discovery of the d-wave HTSCs. De Gennes & Saint-James considered [6] the case of multiple Andreev and normal reflections of quasi-particles in a normal layer (N) of thickness L sandwiched between a superconductor (S) and an insulator (I). Adopting a semiclassical picture, the bound state corresponds to a closed quasi-particle trajectory, and its energy is given by the Bohr–Sommerfeld quantization condition, which requires that the total phase accumulated during one cycle is equal to 2πn (where n is an integer). The total phase accumulated over the closed trajectory (such as that depicted in figure 1) consists of the following two contributions [5]:

1. The phase shift due to an Andreev reflection [7]: −[γ(E) ± Φ], where γ(E) = arccos(E/Δ). Here, the energy E is taken relative to the Fermi level, E_F, Φ is the phase of the superconductor order parameter and the plus (minus) sign corresponds to the conversion of an electron (hole) to a hole (electron). Note also that we consider only quasi-particles within the energy gap, with energy |E| < Δ.

2. The phase accumulated over the trajectory in the normal layer β(E) = 2(k^e– k^h)L/cos(θ), where the momenta of the electron and hole are given by ħk^e,h= [2m(E_F ± E)]^1/2 (assuming a quadratic spectrum for the quasi-particles in the normal metal). Here θ is the angle between the electron trajectory and the normal to the surface (figure 1), and m is the electron mass. In the common case where |E| ≪ E_F, (k^e– k^h) ≅ 2E/ħv_F, where v_F is the Fermi velocity. Therefore, β(E) ≅ 4LE/ħv_F cos(θ).

Applying the Bohr–Sommerfeld quantization condition, the energy of the ABSs can be found by solving

1.1

where the indices correspond to the two different Andreev reflections in the process, as marked in figure 1. In the case of an isotropic (s-wave) superconductor, there are no zero-energy solutions and the ABSs always have finite energy. The situation changes, however, in the case of a d-wave superconductor, where the order parameter is angle-dependent: Δ(θ) = Δ₀ cos[2(θ − α)], where α is the angle between the surface normal and the anti-nodal (lobe) direction (figure 1). In the case where the surface normal is along the nodal direction, as presented in figure 1, α = π/4 and Φ₂− Φ₁ = π, and thus equation (1.1) becomes

1.2

This equation always has a solution for E = 0, for any value of L, because γ(E = 0)₁=γ(E = 0)₂ = π/2 and β(E = 0) = 0.

2. Nodal surface Andreev bound states in the cuprate d-wave superconductors

The ZBCPs featured are very robust in all hole-doped cuprate HTSCs and appeared unexpectedly even when tunnelling in nominal anti-nodal ([100]) or c-axis ([001]) directions. This puzzle was resolved by scanning tunnelling microscopy and spectroscopy (STM and STS) measurements on epitaxial c-axis YBCO films that featured the expected V-shaped gaps on the crystallites, but pronounced ZBCPs on (110) crystallite facets [8]. Surprisingly, nodal facets of only one unit-cell height were found sufficient to host zero-energy ABSs, giving rise to pronounced ZBCPs (figure 2) [8,9]. Another interesting effect regarding the ABSs is their possible penetration into a normal layer deposited on the nodal surface. This question touches upon one of the fundamental properties of the cuprates—the high degree of anisotropy (in particular for the hole-doped HTSCs). This in-ab-plane versus out-of-plane anisotropy manifested itself by a significantly stronger proximity effect (PE), namely longer penetration of superconducting correlations into the normal metal, taking place along the a-axis compared with the c-axis [10]. Moreover, no PE is expected to take place along the nodal direction, e.g. in (110)YBCO/Au junctions. Nevertheless, STS measurements performed by Asulin et al. [11] on such junctions showed that, while there is no evidence for an induced proximity gap in the Au layer, the ABSs clearly penetrate it. Pronounced ZBCPs were measured on Au layers, but their decay with thickness of the Au layer did not conform to the exponential decay typical of ‘conventional’ PE. Rather, their magnitude was more or less constant up to a thickness of 8 nm, with magnitude similar to those detected on the bare superconductor film, but then decayed abruptly with the thickness of the Au film, vanishing above 10 nm. This length scale is shorter than the normal coherence length in our Au films at 4.2 K (greater than 30 nm) and corresponds well to the ballistic mean free path determined by the Au grain size.

Figure 2.

Correlation between tunnelling spectra (at 4.2 K) and the surface morphology of c-axis YCBO film. (a) A 50 × 50 nm² STM topographic image showing crystallites and different crystallographic facets, as indicated. Spectrum (b) was measured on the crystallite, showing the expected V-shaped gap, while (c) was measured on a nodal (110) facet, exhibiting a pronounced ZBCP. The blue curves are fits to the tunnelling theory into a d-wave superconductor. Spectrum (d) was acquired on a step of only one unit-cell height running along the [110] direction, shown in the inset. (Online version in colour.)

Unlike the hole-doped cuprates, ZBCPs were not found in tunnelling spectra measured on the electron-doped cuprates, such as Nd_1.85Ce_0.15CuO_4−y (NCCO) [12] and Pr_2−xCe_xCuO_4−δ (PCCO) [13]. This finding first raised confusion regarding the order-parameter symmetry in the electron-doped cuprates, which were suggested [12] to be predominantly s-wave. However, Dagan et al. [13] showed that PCCO is a weak-coupling ‘dirty’ d-wave superconductor, and the vanishing of the ZBCP can thus be accounted for by the effect of disorder in this material.

ABSs residing on the surface of cuprate HTSCs can provide further, more subtle information about the order-parameter symmetry. One controversial issue that has attracted attention for some time is the possibility of emergence of a state of spontaneous broken time-reversal symmetry at low temperatures (approx. 10 K in YBCO). Such a state is associated with a complex order parameter, either d + is or , as predicted by Fogelström et al. [14] and Laughlin [15], respectively, depending on the direction of the screening currents. Both these order parameters manifest themselves in the tunnelling spectra by split ZBCPs, as was observed by some groups that could also show good fits to the corresponding tunnelling models [16–19]. However, such splitting was either not observed or not interpreted as due to the effects of different competing orders by other groups [20–22].

We now turn to discuss other, more intricate properties that are reflected in the nature of the ABS, beyond the rather well-established d-wave symmetry of the superconducting order parameter in the cuprates and the related anisotropy. In the following, we focus on three main topics: (i) the origin of the 1/8 anomaly in the superconducting dome of lanthanum-based cuprates; (ii) long-ranged PEs in superconducting/ferromagnetic interfaces, i.e. observation of crossed Andreev reflections and/or proximity-induced triplet pairing; and (iii) proximity-induced triplet superconductivity in the doped topological insulator Bi₂Se₃ near the interface with a NbN superconductor.

3. Anti-phase ordering of the d-wave order parameter and the origin of the 1/8 anomaly

The family of lanthanum-based HTSCs, La214, exhibits an anomalous drop of the transition temperature, T_c, at the x = 1/8 doping level (with respect to the ‘unperturbed’ superconducting dome), an effect known as the 1/8 anomaly. At the same doping level, the electronic stripes are known to become static and order commensurately with the underlying lattice (with a periodicity of four unit cells). The coincidence of the two phenomena at x = 1/8 has led to the conjecture that the stripe order competes with superconductivity, and their strong interaction is responsible for this anomaly. This connection, however, has not been fully established yet, neither theoretically nor experimentally. An important step towards understanding the physics of the stripe phase and the corresponding 1/8 anomaly was achieved by Berg et al. [23], suggesting that a state of anti-phase ordering of the order parameter within each CuO₂ plane develops around the x = 1/8 doping level. Apparently, this type of order-parameter modulation within the ab-plane suppresses interlayer Josephson coupling, thus suppressing T_c, while in-plane superconductivity still survives. The local density of states of a phase-biased junction comprising a d-wave superconductor was calculated by Tanaka & Kashiwaya [24], who found that, for π phase difference, zero-energy ABSs form at the interface. Such a junction is analogous to a domain wall (DW) where the order parameter undergoes a π phase shift (termed πDW), described in the anti-phase ordering model put forward by Berg et al. [23]. The implication of such a scenario is that ABSs should appear also in c-axis tunnelling (normal to the ab-plane) and, moreover, that the corresponding ZBCPs should show spatial modulations with a period that depends on the scan direction with respect to the stripes.

Strong evidence for a state of anti-phase ordering was provided by Yuli et al. [25], in a detailed STS investigation performed on (001)La_1.88Sr_0.12CuO₄ (c-axis LSCO, x ≈ 1/8) thin films. Anomalous ZBCPs abundantly appeared in the tunnelling spectra measured on top of c-axis crystallites in these films, while they never appeared (except on (110) facets) for any other doping level. Moreover, spatial modulation of the ZBCP amplitude was observed, with a period that depends on the scan direction (figure 3). The minimal period was found in a direction normal to that where no modulation was observed (α = π/2 in figure 3), and constant ZBCPs were measured along tens of nanometres line scans. The minimal period was approximately 9 nm, corresponding to six stripe distances, which amounts to 24 unit cells of LSCO. In some cases, an asymmetric splitting with a varying degree of imbalance between the negative and positive peak heights was observed. The imbalance ranged from a small difference in peak heights to a state where one of the peaks was completely suppressed. The transition from a nearly fully suppressed positive peak to a nearly fully suppressed negative peak, through an unsplit ZBCP, could take place even within a distance of a few nanometres, as demonstrated in figure 4. Typically, the inversion of the split peak imbalance should take place at the πDW (in the case of strong coupling) owing to the sign change of the order parameter, while far away from the πDW the peaks become symmetric around zero bias again [26].

Figure 3.

(a) Schematic illustration of the one-dimensional anti-phase order. The thick coloured lines represent the domain centres and the DW is represented by the dashed lines. (b) Constant ZBCPs are observed for tunnelling spectra taken in parallel to the DW like the ones connecting points C and D in (a). (c) A modulation of the ZBCP amplitude occurs for any line with α ≠ 0. In the case of the line connecting the points A and B, a double-peak structure emerges because it crosses two DWs. (Online version in colour.)

Figure 4.

Spatial evolution of a split ZBCP corresponding to a πDW at point 2 in the case of strong local pairing. (b) Tunnelling dI/dV versus V spectra exhibiting split ZBCPs and a pronounced centred unsplit ZBCP taken at the indicated positions in (a). The spectra were taken along the 60 nm long line drawn in the topographic map shown in the inset of (b). (Online version in colour.)

4. Long-ranged proximity effects in superconducting/ferromagnetic interfaces

In a conventional superconductor, the Cooper pairs are formed from electrons with an anti-parallel spin alignment and are in the spin-singlet state. By contrast, ferromagnetism favours a parallel alignment of electron spins. Consequently, superconductivity and ferromagnetism are two competing orders that rarely coexist and superconducting order is expected to penetrate a ferromagnet only to very short distances from an interface with a superconductor, typically of the order of ξ_F =  ∼1 nm [27]. This ‘ferromagnetic coherence length’, with D being the diffusivity and E_ex the exchange energy in the ferromagnet, is significantly shorter than the typical penetration depth of superconducting correlations into a non-magnetic normal metal, , which can be a few 100 nm, much larger than the typical superconducting coherence length, ξ_S, of the cuprates. However, long-ranged PE that allows for significant supercurrents was observed in various Josephson junctions comprising a ferromagnetic layer much thicker than ξ_F [28–30]. This long-range PE in ferromagnetic/superconducting (F/S) junctions can be accounted for by the emergence of triplet-pairing superconductivity near the interface, as predicted by various theoretical models [31–33]. One question that is still under debate is regarding the orbital symmetry of the induced triplet state, an issue that cannot be conclusively resolved in Josephson supercurrent experiments. We will show below that STS, and in particular the detection of ZBCPs in the tunnelling spectra, can provide some insights into this problem. Before addressing this problem though, we will show that our Andreev spectroscopic measurements provide strong evidence for another possible mechanism for the long-range PE, which is based on the crossed Andreev reflection effect (CARE) process [34] taking place at the magnetic domain wall. Here, a spin-polarized hole-like quasi-particle coming from one magnetic domain is retro-reflected at the interface as an electron with opposite spin in the adjacent domain, thus overcoming the problem of spin polarization in the ferromagnet. By this, a Cooper pair can then be viewed as effectively being transferred into the F layer. Importantly, the width of the DW should be of the order of (or smaller than) ξ_S in order for the CARE process to occur. This mechanism is thus limited to ferromagnets having narrow DWs, such as SrRuO₃ (SRO), where the DW width is approximately 3 nm [35].

(a). Crossed Andreev reflections in YBCO/SRO interfaces

Results consistent with a CARE were observed in SFS ramp-type junctions with YBCO electrodes (S) and the itinerant ferromagnet SRO [36]. The junctions behaved as typical magnetic tunnelling junctions, as the conductance spectra were always asymmetric, and a few showed bound state peaks at finite bias that shifted with the field. In many of the SFS junctions with a barrier thickness of 10–20 nm, a prominent ZBCP has been observed. This peak was found to decrease linearly with the magnetic field, as expected for Andreev and CARE scattering (figure 5). It was found that the ZBCP height was maximal under zero-field cooling, but decreased significantly at zero field after field cycling. Since under zero-field cooling there are more domains in the SRO barrier than after the field cycling; this finding is in agreement with the CARE mechanism, as illustrated in the inset of figure 5.

Figure 5.

Normalized conductance spectra measured on the YBCO/SRO/YBCO S/F/S junction under zero-field cooling (ZFC) to 4 K, followed by field ramping to 8 T and back to 0 T. Inset: the ZBCP height above the background conductance versus magnetic field. The inset also illustrates schematically the decrease in DW density after field cycling, resulting in a decrease in the ZBCP at zero field (purple dots) compared with that measured directly after ZFC (black squares) (adopted from [36]). (Online version in colour.)

In a related work [37], STS was performed on SRO/(100)YBCO F/S bilayers, revealing narrow strips along which the superconductor order parameter penetrates the ferromagnet to more than 26 nm, an order of magnitude larger than ξ_F. These (parallel) strips were separated by approximately 200 nm from one another, consistent with the known magnetic DW structure in SRO. This behaviour was attributed to CARE taking place in the vicinity of the magnetic DWs as in [36], giving rise to localized long-range PE. Another intriguing effect was observed in studies of ultra-thin SRO layers (nominally 3 nm thick) deposited on the nodal (110)YBCO surface, forming island films. This enabled STS measurements both on the SRO islands and on the bare YBCO surface. While the tunnelling spectra measured far enough from the island's edge (more than a few ξ_S) exhibited the ‘conventional’ ZBCPs, those measured on the islands showed a split ZBCP with an imbalance between peak heights [38]. These data indicate that the splitting occurs at the superconductor side of the interface as a consequence of induced magnetization (see also [39]), confirming theoretical predictions [40], while the imbalance is attributed to the spin polarization in F.

(b). Triplet superconductivity at superconductor/ferromagnet junctions

The above STS investigation was extended to bilayer films made of the half-metallic ferromagnetic La_2/3Ca_1/3MnO₃ (LCMO) grown on two types of high-temperature superconductor films: the hole-doped YBCO [41] and the electron-doped Pr_1.85Ce_­CuO₄ (PCCO) [42]. Surprisingly, the tunnelling spectra revealed long-ranged penetration of the superconductor order parameter into the LCMO layer, to distances as long as 30 nm, more than an order of magnitude larger than the expected coherence length associated with singlet-pairing superconductivity in LCMO, ξ_F < 1 nm. This anomalous PE manifested itself in the tunnelling spectra as gaps and ZBCPs, and in some cases split ZBCPs, that were still detectable even for LCMO/YBCO and LCMO/PCCO bilayers with an LCMO thickness of up to 30 nm.

In contrast to our previous STS study of epitaxial SRO/YBCO bilayers, the long-ranged PE here cannot be attributed to CARE because the magnetic DW widths in LCMO are much larger than the coherence length in both PCCO and LCMO. For comparison, we note that ZBCPs were not observed on the SRO/YBCO bilayers, and thus they must have a different origin than the CARE. Therefore, the superconducting-related features observed on the LCMO, both the gaps and ZBCPs, can most reasonably be accounted for by the emergence of parallel-spin triplet-pairing superconductivity at the bilayer's interfaces, allowing for coexistence of superconductivity and ferromagnetism near the interface and, consequently, for the long-ranged PE. The appearance of ZBCPs (figure 6) suggests that the orbital symmetry of the induced order parameter is anisotropic and sign-changing, of either d-wave or p-wave character, corresponding to an odd or even dependence on the Matsubara frequency, respectively (although odd-frequency s-wave order parameter is also possible; see below). We wish to emphasize here that the fact that ZBCPs were observed also in bilayers comprising PCCO is particularly important, because the tunnelling spectra measured on the bare PCCO never show such features [13,42]. One can thus conclude that the observed ZBCP is strictly associated with the triplet-pairing order parameter and does not just reflect the symmetry of the underlying superconductor. Indeed, Eschrig & Löfwander [32] predicted that the induced order parameter should be predominantly a mixture of s-wave and p-wave components. The split ZBCP features (figure 6), which were observed also on LCMO/PCCO bilayers, suggest the existence of a complex order parameter, e.g. p_x+ ip_y, where the degree of splitting may reflect the strength of the local exchange field [43,44]. This type of order parameter is believed to exist in Sr₂RuO₄, which is conjectured to be a chiral triplet-pairing superconductor [45]. It should be noted here that ZBCPs can also manifest an induced odd-frequency s-wave order parameter, as demonstrated in [46], but the size of the ZBCPs we found along with the observed splitting are more consistent with the above p-wave-related symmetry.

Figure 6.

Tunnelling spectra at 4.2 K measured along a line (100 nm long) on a bilayer of 20 nm thick LCMO deposited on YBCO, showing the splitting of the ZBCP. The upper inset depicts typical proximity gaps on such bilayers. The lower inset portrays the tunnelling configuration. (Online version in colour.)

ZBCPs were found also on the surface of a conventional s-wave superconductor, NbN, proximity coupled to LCMO, indicating induced triplet superconductivity there [47]. The magnitude of the ZBCPs in all three types of junctions discussed here vanished upon applying magnetic fields larger than the saturation field in LCMO [48], manifesting the importance of magnetic inhomogeneity for the emergence of triplet superconductivity at S/F junctions, as predicted [31–33] theoretically.

5. Triplet superconductivity induced in Bi₂Se₃ proximity coupled to NbN

In the search for Majorana modes in proximity-induced topological superconducting junctions, we happened to find a signature of same-spin triplet superconductivity which appears to dominate these elusive elementary excitations [49]. Thin-film junctions and bilayers of the doped topological insulator Bi₂Se₃ and the s-wave superconductor NbN exhibit conductance spectra with coexisting prominent ZBCPs and coherence peaks. Various tunnelling models with different pair potentials such as s-wave, d-wave, d + is and, more importantly, the pair potential of a topological superconductor, which is predicted to give rise to Majorana modes (within vortices) [50], have failed to fit our data. Only the spin-full (equal-spin) triplet p_x+ ip_y pair potential, which breaks time-reversal symmetry, yielded reasonably good fits to our spectra. This conclusion was based on measurements performed on various types of junctions, with different transparencies, two of which are shown in figure 7. Our data thus provide supporting evidence for proximity-induced parallel-spin triplet superconductivity in the Bi₂Se₃ layer near the interface with the NbN film.

Figure 7.

Tunnelling spectra measured on two types of NbN/Bi₂Se₃ tunnel junctions, both showing pronounced ZBCPs and coherence peaks: (a) ‘large’ area (100 × 150 µm²) junction and (b) nanoscale point contact formed by controllably crashing the STM tip into the Bi₂Se₃ film. Good fits to the spectra were obtained using an equal-spin triplet p_x + ip_y pair potential (red curves), whereas other order parameters, such as d-wave (green curve) or that corresponding to a topological superconductor (blue curve), failed to fit our data. (Online version in colour.)

6. Summary

In this article, we reviewed the basics of Andreev bound states at interfaces of superconductors with normal metals, ferromagnets and a topological insulator. Specific experiments were described where the observed ZBCPs were attributed to the sign change of the order parameter such as in the cuprates, induced triplet-pairing superconductivity at YBCO/LCMO and PCCO/LCMO interfaces, and to the crossed Andreev reflection effect in YBCO/SRO junctions. All these effects originate in the PE where the superconductive excitations (the quasi-particles) penetrate into the material adjacent to the superconductor by multiple alternating Andreev and normal reflection processes up to the gap energy. Their penetration length into a ferromagnetic material in contact with the superconductor depends on the spin ordering in this material. Whereas near narrow domain walls CARE can facilitate long-range penetration of superconducting order into a ferromagnet, this mechanism does not apply when the domain walls are wide compared with the coherence length of the superconductor. Here arises the possibility of converting a singlet order parameter as in s- and d-wave superconductors into a same-spin triplet one in the ferromagnet near the interface. This can be allowed in the presence of inhomogeneous magnetization at the interface as occurs near domain walls, even if they are broad compared to the superconductor coherence length, or due to non-collinearity between the S/F interface and F-interior magnetizations. So, to answer the question posed in the title of this article, we can say that, depending on the nature of the material in contact with the superconductor, transport mechanisms of quasi-particles at interfaces can be elucidated from the observed zero-energy ABS, and new induced order-parameter symmetries, as well as evidence for exotic ordering phases in the cuprates, can be found.

Data accessibility

This article has no additional data.

Authors' contributions

O.M. and G.K. jointly prepared this review article, with equal contributions.

Competing interests

We declare we have no competing interests.

Funding

The work of the authors described in this review was supported in part by the Israel Science Foundation, the joint German–Israeli DIP Project, the United States–Israel Binational Science Foundation, the Leverhulme Trust Foundation, the Harry de Jur Chair in Applied Science (O.M.) and the Karl Stoll Chair in advanced materials at the Technion (G.K.).