Scanning-tunneling-microscope observation of the Abrikosov flux lattice and the density of states near and inside a fluxoid

Band-resolved Caroli-de Gennes-Matricon states of multiple-flux-quanta vortices in a multiband superconductor

Superconductors are of type I or II depending on whether they form an Abrikosov vortex lattice. Although bulk lead (Pb) is classified as a prototypical type-I superconductor, we show that its two-band superconductivity allows for single-flux-quantum and multiple-flux-quanta vortices in the intermediate state at millikelvin temperature. Using scanning tunneling microscopy, the winding number of individual vortices is determined from the real space wave function of its Caroli-de Gennes-Matricon bound states. This generalizes the topological index theorem put forward by Volovik for isotropic electronic states to realistic electronic structures. In addition, the bound states due to the two superconducting bands of Pb can be separately detected and the two gaps close independently inside vortices. This yields strong evidence for a low interband coupling.

Superconducting vortices carrying a temperature-dependent fraction of the flux quantum

Magnetic field penetrates type-II bulk superconductors by forming quantum vortices that enclose a magnetic flux equal to the magnetic flux quantum. The flux quantum is a universal quantity that depends only on fundamental constants. Here we investigate isolated vortices in the hole-overdoped Ba_1-xK_xFe₂As₂ (x = 0.77) by using scanning superconducting quantum interference device (SQUID) magnetometry. In many locations, we observed vortices that carried only part of a flux quantum, with a magnitude that varied continuously with temperature. We interpret these features as quantum vortices with non-universally quantized (fractional) magnetic flux whose magnitude is determined by the temperature-dependent parameters of a multiband superconductor. The demonstrated mobility and manipulability of the fractional vortices may enable applications in fluxonics-based computing.

A spectroscopic-imaging scanning tunneling microscope in vector magnetic field

Cryogenic scanning tunneling microscopy and spectroscopy (STM/STS) performed in a high vector magnetic field provide unique possibilities for imaging surface magnetic structures and anisotropic superconductivity and exploring spin physics in quantum materials with atomic precision. Here, we describe the design, construction, and performance of a low-temperature, ultra-high-vacuum (UHV) spectroscopic-imaging STM equipped with a vector magnet capable of applying a field of up to 3 T in any direction with respect to the sample surface. The STM head is housed in a fully bakeable UHV compatible cryogenic insert and is operational over variable temperatures ranging from ∼300 down to 1.5 K. The insert can be easily upgraded using our home-designed ³He refrigerator. In addition to layered compounds, which can be cleaved at a temperature of either ∼300, ∼77, or ∼4.2 K to expose an atomically flat surface, thin films can also be studied by directly transferring using a UHV suitcase from our oxide thin-film laboratory. Samples can be treated further with a heater and a liquid helium/nitrogen cooling stage on a three-axis manipulator. The STM tips can be treated in vacuo by e-beam bombardment and ion sputtering. We demonstrate the successful operation of the STM with varying the magnetic field direction. Our facility provides a way to study materials in which magnetic anisotropy is a key factor in determining the electronic properties such as in topological semimetals and superconductors.

The magnetic field driven superconductor-metal transition in disordered hole-overdoped cuprates

By solving the Bogoliubov-de Gennes equations for ad-wave superconductor, we explore how the interplay between disorder and the orbital depairing of an external magnetic field influences the superconductor-metal transition of the hole-overdoped cuprates. For highly disordered systems, we find granular Cooper paring to persist above the critical field where the superfluid stiffness goes to zero. We also show that because the vortices are attracted to regions where the superconducting pairing is already weak, the Caroli-de Gennes-Matricon zero-bias peak in the local density of states at the vortex cores disappears already at moderate disorder.

Bamboo-like Vortex Chains Confined in Canals with Suppressed Superconductivity and Standing Waves of Quasiparticles

The vortex core can be regarded as a nanoscale confined system for quasiparticles in a type-II superconductor. It is very interesting to investigate the interplay between the vortex core and other microscopic quantum confined systems. We observe band-like canals with the width of about 15 nm on the surface of KCa₂(Fe_1-xNi_x)₄As₄F₂ (x = 0.05) by scanning tunneling microscopy. Some canals suppress superconductivity and confine parallel standing waves due to the quasiparticle interference. Upon magnetic fields being applied, some elongated vortices are formed within canals showing bamboo-like one-dimensional vortex chains. Interestingly, the confined vortex cores are elongated roughly along the perpendicular direction of canals, and the local density of states at positive and negative energies shows an in-phase oscillation at zero field; but, it becomes out-of-phase crossing the vortex cores. Our work reveals a new type of vortex patterns in confined canals and its interplay with the quasiparticle interference.

A sacrificial magnet concept for field dependent surface science studies

We demonstrate a straightforward approach to integrating a magnetic field into a low-temperature scanning tunneling microscope (STM) by adhering an NdFeB permanent magnet to a magnetizable sample plate. To render our magnet concept compatible with high-temperature sample cleaning procedures, we make the irreversible demagnetization of the magnet a central part of our preparation cycle. After sacrificing the magnet by heating it above its Curie temperature, we use a transfer tool to attach a new magnet in-situ prior to transferring the sample into the STM. We characterize the magnetic field created by the magnet using the Abrikosov vortex lattice of superconducting NbSe_2. Excellent agreement between the distance dependent magnetic fields from experiments and simulations allows us to predict the magnitude and orientation of magnetic flux at any location with respect to the magnet and the sample plate. Our concept is an accessible solution for field-dependent surface science studies that require fields in the range of up to 400 mT and otherwise detrimental heating procedures.•Accessible magnetic field generation.•Selectable field strength and orientation.•Compatible with high-temperature sample preparation.

InAs/MoRe Hybrid Semiconductor/Superconductor Nanowire Devices

Implementing superconductors capable of proximity-inducing a large energy gap in semiconductors in the presence of strong magnetic fields is a major goal toward applications of semiconductor/superconductor hybrid materials in future quantum information technologies. Here, we study the performance of devices consisting of InAs nanowires in electrical contact with molybdenum-rhenium (MoRe) superconducting alloys. The MoRe thin films exhibit transition temperatures of ∼10 K and critical fields exceeding 6 T. Normal/superconductor devices enabled tunnel spectroscopy of the corresponding induced superconductivity, which was maintained up to ∼10 K, and MoRe-based Josephson devices exhibited supercurrents and multiple Andreev reflections. We determine an induced superconducting gap lower than expected from the transition temperature and observe gap softening at finite magnetic field. These may be common features for hybrids based on large-gap, type II superconductors. The results encourage further development of MoRe-based hybrids.

Electronic gap characterization at mesoscopic scale via scanning probe microscopy under ambient conditions

Electronic gaps play an important role in the electric and optical properties of materials. Although various experimental techniques, such as scanning tunnelling spectroscopy and optical or photoemission spectroscopy, are normally used to perform electronic band structure characterizations, it is still challenging to measure the electronic gap at the nanoscale under ambient conditions. Here we report a scanning probe microscopic technique to characterize the electronic gap with nanometre resolution at room temperature and ambient pressure. The technique probes the electronic gap by monitoring the changes of the local quantum capacitance via the Coulomb force at a mesoscopic scale. We showcase this technique by characterizing several 2D semiconductors and van der Waals heterostructures under ambient conditions.

Superconducting Materials and Devices Grown by Focused Ion and Electron Beam Induced Deposition

Since its discovery in 1911, superconductivity has represented an equally inciting and fascinating field of study in several areas of physics and materials science, ranging from its most fundamental theoretical understanding, to its practical application in different areas of engineering. The fabrication of superconducting materials can be downsized to the nanoscale by means of Focused Ion/Electron Beam Induced Deposition: nanopatterning techniques that make use of a focused beam of ions or electrons to decompose a gaseous precursor in a single step. Overcoming the need to use a resist, these approaches allow for targeted, highly-flexible nanopatterning of nanostructures with lateral resolution in the range of 10 nm to 30 nm. In this review, the fundamentals of these nanofabrication techniques are presented, followed by a literature revision on the published work that makes use of them to grow superconducting materials, the most remarkable of which are based on tungsten, niobium, molybdenum, carbon, and lead. Several examples of the application of these materials to functional devices are presented, related to the superconducting proximity effect, vortex dynamics, electric-field effect, and to the nanofabrication of Josephson junctions and nanoSQUIDs. Owing to the patterning flexibility they offer, both of these techniques represent a powerful and convenient approach towards both fundamental and applied research in superconductivity.

Millikelvin scanning tunneling microscope at 20/22 T with a graphite enabled stick-slip approach and an energy resolution below 8 μeV: Application to conductance quantization at 20 T in single atom point contacts of Al and Au and to the charge density wave of 2H-NbSe2

We describe a scanning tunneling microscope (STM) that operates at magnetic fields up to 22 T and temperatures down to 80 mK. We discuss the design of the STM head, with an improved coarse approach, the vibration isolation system, and efforts to improve the energy resolution using compact filters for multiple lines. We measure the superconducting gap and Josephson effect in aluminum and show that we can resolve features in the density of states as small as 8 μeV. We measure the quantization of conductance in atomic size contacts and make atomic resolution and density of states images in the layered material 2H-NbSe₂. The latter experiments are performed by continuously operating the STM at magnetic fields of 20 T in periods of several days without interruption.

Coherent coupling between vortex bound states and magnetic impurities in 2D layered superconductors

Bound states in superconductors are expected to exhibit a spatially resolved electron-hole asymmetry which is the hallmark of their quantum nature. This asymmetry manifests as oscillations at the Fermi wavelength, which is usually tiny and thus washed out by thermal broadening or by scattering at defects. Here we demonstrate theoretically and confirm experimentally that, when coupled to magnetic impurities, bound states in a vortex core exhibit an emergent axial electron-hole asymmetry on a much longer scale, set by the coherence length. We study vortices in 2H-NbSe2 and in 2H-NbSe1.8S0.2 with magnetic impurities, characterizing these with detailed Hubbard-corrected density functional calculations. We find that the induced electron-hole imbalance depends on the band character of the superconducting material. Our results show that coupling between quantum bound states in superconductors is remarkably robust and has a strong influence in tunneling measurements.

Friedel Oscillations of Vortex Bound States under Extreme Quantum Limit in KCa_{2}Fe_{4}As_{4}F_{2}

We report the observation of discrete bound states with the energy levels deviating from the widely believed ratio of 1∶3∶5 in the vortices of an iron-based superconductor KCa_{2}Fe_{4}As_{4}F_{2} through scanning tunneling microscopy (STM). Meanwhile Friedel oscillations of vortex bound states are also observed for the first time in related vortices. By doing self-consistent calculations of Bogoliubov-de Gennes equations, we find that at extreme quantum limit, the superconducting order parameter exhibits a Friedel-like oscillation, which modifies the energy levels of the vortex bound states and explains why it deviates from the ratio of 1∶3∶5. The observed Friedel oscillations of the bound states can also be roughly interpreted by the theoretical calculations, however some features at high energies could not be explained. We attribute this discrepancy to the high energy bound states with the influence of nearby impurities. Our combined STM measurement and the self-consistent calculations illustrate a generalized feature of vortex bound states in type-II superconductors.

Electronic and Magnetic Characterization of Epitaxial CrBr3 Monolayers on a Superconducting Substrate

The ability to imprint a given material property to another through a proximity effect in layered 2D materials has opened the way to the creation of designer materials. Here, molecular-beam epitaxy is used for direct synthesis of a superconductor-ferromagnet heterostructure by combining superconducting niobium diselenide (NbSe₂ ) with the monolayer ferromagnetic chromium tribromide (CrBr₃ ). Using different characterization techniques and density-functional theory calculations, it is confirmed that the CrBr₃ monolayer retains its ferromagnetic ordering with a magnetocrystalline anisotropy favoring an out-of-plane spin orientation. Low-temperature scanning tunneling microscopy measurements show a slight reduction of the superconducting gap of NbSe₂ and the formation of a vortex lattice on the CrBr₃ layer in experiments under an external magnetic field. The results contribute to the broader framework of exploiting proximity effects to realize novel phenomena in 2D heterostructures.

Competing Energy Scales in Topological Superconducting Heterostructures

Artificially engineered topological superconductivity has emerged as a viable route to create Majorana modes. In this context, proximity-induced superconductivity in materials with a sizable spin-orbit coupling has been intensively investigated in recent years. Although there is convincing evidence that superconductivity may indeed be induced, it has been difficult to elucidate its topological nature. Here, we engineer an artificial topological superconductor by progressively introducing superconductivity (Nb), strong spin-orbital coupling (Pt), and topological states (Bi₂Te₃). Through spectroscopic imaging of superconducting vortices within the bare s-wave superconducting Nb and within proximitized Pt and Bi₂Te₃ layers, we detect the emergence of a zero-bias peak that is directly linked to the presence of topological surface states. Our results are rationalized in terms of competing energy trends which are found to impose an upper limit to the size of the minigap separating Majorana and trivial modes, its size being ultimately linked to fundamental materials properties.

Observation of Distinct Spatial Distributions of the Zero and Nonzero Energy Vortex Modes in (Li_{0.84}Fe_{0.16})OHFeSe

The energy and spatial distributions of vortex bound state in superconductors carry important information about superconducting pairing and the electronic structure. Although discrete vortex states, and sometimes a zero energy mode, had been observed in several iron-based superconductors, their spatial properties are rarely explored. In this study, we used low-temperature scanning tunneling microscopy to measure the vortex state of (Li,Fe)OHFeSe with high spatial resolution. We found that the nonzero energy states display clear spatial oscillations with a period corresponding to bulk Fermi wavelength; while in contrast, the zero energy mode does not show such oscillation, which suggests its distinct electronic origin. Furthermore, the oscillations of positive and negative energy states near E_{F} are found to be clearly out of phase. Based on a two-band model calculation, we show that our observation is more consistent with an s_{++} wave pairing in the bulk of (Li, Fe)OHFeSe, and superconducting topological states on the surface.

On the Remarkable Superconductivity of FeSe and Its Close Cousins

Emergent electronic phenomena in iron-based superconductors have been at the forefront of condensed matter physics for more than a decade. Much has been learned about the origins and intertwined roles of ordered phases, including nematicity, magnetism, and superconductivity, in this fascinating class of materials. In recent years, focus has been centered on the peculiar and highly unusual properties of FeSe and its close cousins. This family of materials has attracted considerable attention due to the discovery of unexpected superconducting gap structures, a wide range of superconducting critical temperatures, and evidence for nontrivial band topology, including associated spin-helical surface states and vortex-induced Majorana bound states. Here, we review superconductivity in iron chalcogenide superconductors, including bulk FeSe, doped bulk FeSe, FeTe1−xSex, intercalated FeSe materials, and monolayer FeSe and FeTe1−xSex on SrTiO3. We focus on the superconducting properties, including a survey of the relevant experimental studies, and a discussion of the different proposed theoretical pairing scenarios. In the last part of the paper, we review the growing recent evidence for nontrivial topological effects in FeSe-related materials, focusing again on interesting implications for superconductivity.

Intra- and intermolecular self-assembly of a 20-nm-wide supramolecular hexagonal grid

For the past three decades, the coordination-driven self-assembly of three-dimensional structures has undergone rapid progress; however, parallel efforts to create large discrete two-dimensional architectures-as opposed to polymers-have met with limited success. The synthesis of metallo-supramolecular systems with well-defined shapes and sizes in the range of 10-100 nm remains challenging. Here we report the construction of a series of giant supramolecular hexagonal grids, with diameters on the order of 20 nm and molecular weights greater than 65 kDa, through a combination of intra- and intermolecular metal-mediated self-assembly steps. The hexagonal intermediates and the resulting self-assembled grid architectures were imaged at submolecular resolution by scanning tunnelling microscopy. Characterization (including by scanning tunnelling spectroscopy) enabled the unambiguous atomic-scale determination of fourteen hexagonal grid isomers.

Synthesis of Metallopolymers and Direct Visualization of the Single Polymer Chain

During the past few decades, the study of the single polymer chain has attracted considerable attention with the goal of exploring the structure-property relationship of polymers. It still, however, remains challenging due to the variability and low atomic resolution of the amorphous single polymer chain. Here, we demonstrated a new strategy to visualize the single metallopolymer chain with a hexameric or trimeric supramolecule as a repeat unit, in which Ru(II) with strong coordination and Fe(II) with weak coordination were combined together in a stepwise manner. With the help of ultrahigh-vacuum, low-temperature scanning tunneling microscopy (UHV-LT-STM) and scanning tunneling spectroscopy (STS), we were able to directly visualize both Ru(II) and Fe(II), which act as staining reagents on the repeat units, thus providing detailed structural information for the single polymer chain. As such, the direct visualization of the single random polymer chain is realized to enhance the characterization of polymers at the single-molecule level.

Observation of Discrete Conventional Caroli-de Gennes-Matricon States in the Vortex Core of Single-Layer FeSe/SrTiO_{3}

Using low-temperature scanning tunneling microscopy (STM), we studied the vortex states of single-layer FeSe film on a SrTiO_{3} (100) substrate, and the local behaviors of superconductivity at sample boundaries. We clearly observed multiple discrete Caroli-de Gennes-Matricon states in the vortex core, and quantitative analysis shows their energies well follow the formula: E=μΔ^{2}/E_{F}, where μ is a half integer (±1/2,±3/2,±5/2…) and Δ is the mean superconducting gap over the Fermi surface. Meanwhile, a fully gapped spectrum without states near zero bias is observed at the [110]_{Fe} oriented boundary of 1 and 2 ML FeSe films, and atomic step edge of 1 ML FeSe. Accompanied with theoretical calculations, our results indicate an s-wave pairing without sign change in the high-T_{C} FeSe/SrTiO_{3} superconductor.

Relation of Superconducting Pairing Symmetry and Non-Magnetic Impurity Effects in Vortex States

Non-magnetic impurity scattering effects on the vortex core states are theoretically studied to clarify the contributions from the sign-change of the pairing function in anisotropic superconductors. The vortex states are calculated by the Eilenberger theory in superconductors with p x -wave pairing symmetry, as well as the corresponding anisotropic s-wave symmetry. From the spatial structure of the pair potential and the local electronic states around a vortex, we examine the differences between anisotropic superconductors with and without sign-change of the pairing function, and estimate how twofold symmetric vortex core images change with increasing the impurity scattering rate both in the Born and the unitary limits. We found that twofold symmetric vortex core image of zero-energy local density of states changes the orientation of the twofold symmetry with increasing the scattering rate when the sign change occurs in the pairing function. Without the sign change, the vortex core shape reduces to circular one with approaching dirty cases. These results of the impurity effects are valuable for identifying the pairing symmetry by observation of the vortex core image by the STM observation.

Local Josephson vortex generation and manipulation with a Magnetic Force Microscope

Josephson vortices play an essential role in superconducting quantum electronics devices. Often seen as purely conceptual topological objects, 2π-phase singularities, their observation and manipulation are challenging. Here we show that in Superconductor-Normal metal-Superconductor lateral junctions Josephson vortices have a peculiar magnetic fingerprint that we reveal in Magnetic Force Microscopy (MFM) experiments. Based on this discovery, we demonstrate the possibility of the Josephson vortex generation and manipulation by the magnetic tip of a MFM, thus paving a way for the remote inspection and control of individual nano-components of superconducting quantum circuits.

Transport evidence of asymmetric spin-orbit coupling in few-layer superconducting 1Td-MoTe2

Two-dimensional transition metal dichalcogenides MX₂ (M = W, Mo, Nb, and X = Te, Se, S) with strong spin-orbit coupling possess plenty of novel physics including superconductivity. Due to the Ising spin-orbit coupling, monolayer NbSe₂ and gated MoS₂ of 2H structure can realize the Ising superconductivity, which manifests itself with in-plane upper critical field far exceeding Pauli paramagnetic limit. Surprisingly, we find that a few-layer 1T_d structure MoTe₂ also exhibits an in-plane upper critical field which goes beyond the Pauli paramagnetic limit. Importantly, the in-plane upper critical field shows an emergent two-fold symmetry which is different from the isotropic in-plane upper critical field in 2H transition metal dichalcogenides. We show that this is a result of an asymmetric spin-orbit coupling in 1T_d transition metal dichalcogenides. Our work provides transport evidence of a new type of asymmetric spin-orbit coupling in transition metal dichalcogenides which may give rise to novel superconducting and spin transport properties.

Evidence of nematic order and nodal superconducting gap along [110] direction in RbFe2As2

Unconventional superconductivity often intertwines with various forms of order, such as the nematic order which breaks the rotational symmetry of the lattice. Here we report a scanning tunneling microscopy study on RbFe₂As₂, a heavily hole-doped Fe-based superconductor (FeSC). We observe significant symmetry breaking in its electronic structure and magnetic vortex which differentiates the (π, π) and (π, -π) directions of the unfolded Brillouin zone. It is thus a novel nematic state, distinct from the nematicity of undoped/lightly-doped FeSCs which breaks the (π, 0)/(0, π) equivalence. Moreover, we observe a clear V-shaped superconducting gap. The gap is suppressed on surface Rb vacancies and step edges, and the suppression is particularly strong at the [110]-oriented edges. This is possibly due to a \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{d}}_{{{x}}^2 - {{y}}^2}$$\end{document}dx2-y2 like pairing component with nodes along the [110] directions. Our results thus highlight the intimate connection between nematicity and superconducting pairing in iron-based superconductors.

Exotic electronic order may emerge and intertwine with superconductivity in iron-based superconductors. Here, the authors observe asymmetric electronic and superconducting gap structure in heavily hole-doped RbFe₂As₂ along the (π, π) and (π, -π) directions in reciprocal space, suggesting a novel (π, π) nematic phase emerging.

Half-quantum vortices and walls bounded by strings in the polar-distorted phases of topological superfluid 3He

Symmetries of the physical world have guided formulation of fundamental laws, including relativistic quantum field theory and understanding of possible states of matter. Topological defects (TDs) often control the universal behavior of macroscopic quantum systems, while topology and broken symmetries determine allowed TDs. Taking advantage of the symmetry-breaking patterns in the phase diagram of nanoconfined superfluid 3He, we show that half-quantum vortices (HQVs)-linear topological defects carrying half quantum of circulation-survive transitions from the polar phase to other superfluid phases with polar distortion. In the polar-distorted A phase, HQV cores in 2D systems should harbor non-Abelian Majorana modes. In the polar-distorted B phase, HQVs form composite defects-walls bounded by strings hypothesized decades ago in cosmology. Our experiments establish the superfluid phases of 3He in nanostructured confinement as a promising topological media for further investigations ranging from topological quantum computing to cosmology and grand unification scenarios.

Tunneling into the Vortex State of NbSe2 with van der Waals Junctions

We have performed device-based tunneling spectroscopy of NbSe₂ in the vortex state with a magnetic field applied both parallel and perpendicular to the a- b plane. Our devices consist of layered semiconductors placed on top of exfoliated NbSe₂ using the van der Waals transfer technique. At zero field, the spectrum exhibits a hard gap, and the quasiparticle peak is split into low- and high-energy features. The two features, associated with the effective two-band nature of superconductivity in NbSe₂, exhibit markedly distinct responses to the application of magnetic field, suggesting an order-of-magnitude difference in the spatial extent of the vortex cores of the two bands. At energies below the superconducting gap, the hard gap gives way to vortex-bound Caroli-de Gennes-Matricon states, allowing the detection of individual vortices as they enter and exit the junction. Analysis of the subgap spectra upon application of parallel magnetic field allows us to track the process of vortex surface formation and spatial rearrangement in the bulk.

Scanning Tunneling Microscopy of an Air Sensitive Dichalcogenide Through an Encapsulating Layer

Many atomically thin exfoliated two-dimensional (2D) materials degrade when exposed to ambient conditions. They can be protected and investigated by means of transport and optical measurements if they are encapsulated between chemically inert single layers in the controlled atmosphere of a glovebox. Here, we demonstrate that the same encapsulation procedure is also compatible with scanning tunneling microscopy (STM) and spectroscopy (STS). To this end, we report a systematic STM/STS investigation of a model system consisting of an exfoliated 2H-NbSe₂ crystal capped with a protective 2H-MoS₂ monolayer. We observe different electronic coupling between MoS₂ and NbSe₂ from a strong coupling when their lattices are aligned within a few degrees to essentially no coupling for 30° misaligned layers. We show that STM always probes intrinsic NbSe₂ properties such as the superconducting gap and charge density wave at low temperature when setting the tunneling bias inside the MoS₂ band gap, irrespective of the relative angle between the NbSe₂ and MoS₂ lattices. This study demonstrates that encapsulation is fully compatible with STM/STS investigations of 2D materials.

Andreev bound states at spin-active interfaces

Andreev bound states are ubiquitous in superconducting hybrid structures. They are formed near impurities, in Josephson junctions, in vortex cores and at interfaces. At spin-active superconductor-ferromagnet interfaces, Andreev bound states are formed due to spin-dependent scattering phases. Spin-dependent phase shifts are an important ingredient for the generation of triplet Cooper pairs in superconductor-ferromagnet hybrid structures. Spectroscopy of Andreev bound states is a powerful probe of superconducting order parameter symmetry, as well as spin-dependent interface scattering and the triplet proximity effect.This article is part of the theme issue 'Andreev bound states'.

Multifaceted properties of Andreev bound states: interplay of symmetry and topology

Andreev bound states (ABSs) ubiquitously emerge as a consequence of non-trivial topological structures of the order parameter of superfluids and superconductors and significantly contribute to thermodynamics and low-energy quantum transport phenomena. We here share the current status of our knowledge on their multifaceted properties such as Majorana fermions and odd-frequency pairing. A unified concept behind ABSs originates from a soliton state in the one-dimensional Dirac equation with mass domain wall and interplay of ABSs with symmetry and topology enrich their physical characteristics. We make an overview of ABSs with a special focus on superfluid 3He. The quantum liquid confined to restricted geometries serves as a rich repository of noteworthy quantum phenomena, such as the mass acquisition of Majorana fermions driven by spontaneous symmetry breaking, topological quantum criticality, Weyl superfluidity and the anomalous magnetic response. The marriage of the superfluid 3He and nano-fabrication techniques will take one to a new horizon of topological quantum phenomena associated with ABSs.This article is part of the theme issue 'Andreev bound states'.

Expansion of a superconducting vortex core into a diffusive metal

Vortices in quantum condensates exist owing to a macroscopic phase coherence. Here we show, both experimentally and theoretically, that a quantum vortex with a well-defined core can exist in a rather thick normal metal, proximized with a superconductor. Using scanning tunneling spectroscopy we reveal a proximity vortex lattice at the surface of 50 nm-thick Cu-layer deposited on Nb. We demonstrate that these vortices have regular round cores in the centers of which the proximity minigap vanishes. The cores are found to be significantly larger than the Abrikosov vortex cores in Nb, which is related to the effective coherence length in the proximity region. We develop a theoretical approach that provides a fully self-consistent picture of the evolution of the vortex with the distance from Cu/Nb interface, the interface impedance, applied magnetic field, and temperature. Our work opens a way for the accurate tuning of the superconducting properties of quantum hybrids.

Discrete energy levels of Caroli-de Gennes-Matricon states in quantum limit in FeTe0.55Se0.45

Caroli-de Gennes-Matricon (CdGM) states were predicted in 1964 as low-energy excitations within vortex cores of type-II superconductors. In the quantum limit, the energy levels of these states were predicted to be discrete with the basic levels at ±μΔ2/EF (μ = 1/2, 3/2, 5/2, …) with Δ the superconducting energy gap and EF the Fermi energy. However, due to the small ratio of Δ/EF in most type-II superconductors, it is very difficult to observe the discrete CdGM states, but rather a symmetric peak which appears at zero bias at the vortex center. Here we report the clear observation of these discrete energy levels of CdGM states in FeTe0.55Se0.45. The rather stable energies of these bound state peaks vs. space clearly validate our conclusion. Analysis based on the energies of these CdGM states indicates that the Fermi energy in the present system is very small.

Orphan Spins in the S=5/2 Antiferromagnet CaFe_{2}O_{4}

CaFe_{2}O_{4} is an anisotropic S=5/2 antiferromagnet with two competing A (↑↑↓↓) and B (↑↓↑↓) magnetic order parameters separated by static antiphase boundaries at low temperatures. Neutron diffraction and bulk susceptibility measurements, show that the spins near these boundaries are weakly correlated and a carry an uncompensated ferromagnetic moment that can be tuned with a magnetic field. Spectroscopic measurements find these spins are bound with excitation energies less than the bulk magnetic spin waves and resemble the spectra from isolated spin clusters. Localized bound orphaned spins separate the two competing magnetic order parameters in CaFe_{2}O_{4}.

Observation of Caroli-de Gennes-Matricon Vortex States in YBa_{2}Cu_{3}O_{7-δ}

The copper oxides present the highest superconducting temperature and properties at odds with other compounds, suggestive of a fundamentally different superconductivity. In particular, the Abrikosov vortices fail to exhibit localized states expected and observed in all clean superconductors. We have explored the possibility that the elusive vortex-core signatures are actually present but weak. Combining local tunneling measurements with large-scale theoretical modeling, we positively identify the vortex states in YBa_{2}Cu_{3}O_{7-δ}. We explain their spectrum and the observed variations thereof from one vortex to the next by considering the effects of nearby vortices and disorder in the vortex lattice. We argue that the superconductivity of copper oxides is conventional, but the spectroscopic signature does not look so because the superconducting carriers are a minority.

Conformal Vortex Crystals

We investigate theoretically globally nonuniform configurations of quantized-flux vortices in clean superconductors trapped by an external force field that induces a nonuniform vortex density profile. Using an extensive series of numerical simulations, we demonstrate that, for suitable choices of the force field, and bellow a certain transition temperature, the vortex system self-organizes into highly inhomogeneous conformal crystals in a way as to minimize the total energy. These nonuniform structures are topologically ordered and can be mathematically mapped into a triangular Abrikosov lattice via a conformal transformation. Above the crystallization temperature, the conformal vortex crystal becomes unstable and gives place to a nonuniform polycrystalline structure. We propose a simple method to engineer the potential energy profile necessary for the observation of conformal crystals of vortices, which can also be applied to other 2D particle systems, and suggest possible experiments in which conformal or quasi-conformal vortex crystals could be observed in bulk superconductors and in thin films.

Tunable odd-frequency triplet pairing states and skyrmion modes in chiral p-wave superconductor

Bogliubov-de Gennes equations are solved self-consistently to investigate the properties of bound states in chiral p-wave superconductive disks. It shows that either an s-wave or the mixed d- and s-wave state with odd-frequency and spin-triplet symmetry is induced at the vortex core, depending both on the chirality of the pairing states and on the vortex topology. It is also found that the odd-frequency triplet even parity (OTE) bound state can be manipulated with a local non-magnetic potential. Interestingly, with an appropriate potential amplitude, the zero-energy OTE bound state can be stabilized at a distance from the vortex core and from the local potential. Possible existences of the Majorana fermion modes are expected if the particle-hole symmetry property is applied to the zero-energy OTE bound state. Moreover, skyrmion modes with an integer topological charge have been found to exist.

Quantum fluctuations and stability of vortex lattices in a nonresonantly pumped exciton-polariton condensate

The dynamics of an exciton-polariton condensate (EPC) subject to harmonic confinement can cause spontaneously formed vortices to arrange into a triangular vortex lattice. The stability of such a spontaneously formed vortex lattice is still unknown. We investigate the quantum fluctuations of vortex lattices in a rapidly rotating EPC with a rotation frequency close to the harmonic trap. In such a large condensate, we find that a vortex lattice with a triangular structure is stable, whereas one with a square structure becomes unstable. This result indicates that a driven-dissipative vortex array with strong quantum fluctuations can occur in an EPC.

Observation of superconducting vortex clusters in S/F hybrids

While Abrikosov vortices repel each other and form a uniform vortex lattice in bulk type-II superconductors, strong confinement potential profoundly affects their spatial distribution eventually leading to vortex cluster formation. The confinement could be induced by the geometric boundaries in mesoscopic-size superconductors or by the spatial modulation of the magnetic field in superconductor/ferromagnet (S/F) hybrids. Here we study the vortex confinement in S/F thin film heterostructures and we observe that vortex clusters appear near magnetization inhomogeneities in the ferromagnet, called bifurcations. We use magnetic force microscopy to image magnetic bifurcations and superconducting vortices, while high resolution scanning tunneling microscopy is used to obtain detailed information of the local electronic density of states outside and inside the vortex cluster. We find an intervortex spacing at the bifurcation shorter than the one predicted for the same superconductor in a uniform magnetic field equal to the thermodynamical upper critical field H_c2. This result is due to a local enhanced stray field and a competition between vortex-vortex repulsion and Lorentz force. Our findings suggest that special magnetic topologies could result in S/F hybrids that support superconductivity even when locally the vortex density exceeds the thermodynamic critical threshold value beyond which the superconductivity is destroyed.

Superconducting topological surface states in the noncentrosymmetric bulk superconductor PbTaSe2

The search for topological superconductors (TSCs) is one of the most urgent contemporary problems in condensed matter systems. TSCs are characterized by a full superconducting gap in the bulk and topologically protected gapless surface (or edge) states. Within each vortex core of TSCs, there exists the zero-energy Majorana bound states, which are predicted to exhibit non-Abelian statistics and to form the basis of the fault-tolerant quantum computation. To date, no stoichiometric bulk material exhibits the required topological surface states (TSSs) at the Fermi level (EF) combined with fully gapped bulk superconductivity. We report atomic-scale visualization of the TSSs of the noncentrosymmetric fully gapped superconductor PbTaSe2. Using quasi-particle scattering interference imaging, we find two TSSs with a Dirac point at E ≅ 1.0 eV, of which the inner TSS and the partial outer TSS cross EF, on the Pb-terminated surface of this fully gapped superconductor. This discovery reveals PbTaSe2 as a promising candidate for TSC.

Physical Realization of von Neumann Lattices in Rotating Bose Gases with Dipole Interatomic Interactions

This paper reports a novel type of vortex lattice, referred to as a bubble crystal, which was discovered in rapidly rotating Bose gases with long-range interactions. Bubble crystals differ from vortex lattices which possess a single quantum flux per unit cell, while atoms in bubble crystals are clustered periodically and surrounded by vortices. No existing model is able to describe the vortex structure of bubble crystals; however, we identified a mathematical lattice, which is a subset of coherent states and exists periodically in the physical space. This lattice is called a von Neumann lattice, and when it possesses a single vortex per unit cell, it presents the same geometrical structure as an Abrikosov lattice. In this report, we extend the von Neumann lattice to one with an integral number of flux quanta per unit cell and demonstrate that von Neumann lattices well reproduce the translational properties of bubble crystals. Numerical simulations confirm that, as a generalized vortex, a von Neumann lattice can be physically realized using vortex lattices in rapidly rotating Bose gases with dipole interatomic interactions.

Quantitative nanoscale vortex imaging using a cryogenic quantum magnetometer

Microscopic studies of superconductors and their vortices play a pivotal role in understanding the mechanisms underlying superconductivity. Local measurements of penetration depths or magnetic stray fields enable access to fundamental aspects such as nanoscale variations in superfluid densities or the order parameter symmetry of superconductors. However, experimental tools that offer quantitative, nanoscale magnetometry and operate over large ranges of temperature and magnetic fields are still lacking. Here, we demonstrate the first operation of a cryogenic scanning quantum sensor in the form of a single nitrogen-vacancy electronic spin in diamond, which is capable of overcoming these existing limitations. To demonstrate the power of our approach, we perform quantitative, nanoscale magnetic imaging of Pearl vortices in the cuprate superconductor YBa2Cu3O7-δ. With a sensor-to-sample distance of ∼10 nm, we observe striking deviations from the prevalent monopole approximation in our vortex stray-field images, and find excellent quantitative agreement with Pearl's analytic model. Our experiments provide a non-invasive and unambiguous determination of the system's local penetration depth and are readily extended to higher temperatures and magnetic fields. These results demonstrate the potential of quantitative quantum sensors in benchmarking microscopic models of complex electronic systems and open the door for further exploration of strongly correlated electron physics using scanning nitrogen-vacancy magnetometry.

Hubbard Model for Atomic Impurities Bound by the Vortex Lattice of a Rotating Bose-Einstein Condensate

We investigate cold bosonic impurity atoms trapped in a vortex lattice formed by condensed bosons of another species. We describe the dynamics of the impurities by a bosonic Hubbard model containing occupation-dependent parameters to capture the effects of strong impurity-impurity interactions. These include both a repulsive direct interaction and an attractive effective interaction mediated by the Bose-Einstein condensate. The occupation dependence of these two competing interactions drastically affects the Hubbard model phase diagram, including causing the disappearance of some Mott lobes.

Engineering topological superconductors using surface atomic-layer/molecule hybrid materials

Surface atomic-layer (SAL) superconductors consisting of epitaxially grown metal adatoms on a clean semiconductor surface have been recently established. Compared to conventional metal thin films, they have two important features: (i) space-inversion symmetry-breaking throughout the system and (ii) high sensitivity to surface adsorption of foreign species. These potentially lead to manifestation of the Rashba effect and a Zeeman field exerted by adsorbed magnetic organic molecules. After introduction of the archetypical SAL superconductor Si(111)-(√7 × √3)-In, we describe how these features are utilized to engineer a topological superconductor with Majorana fermions and discuss its promises and expected challenges.

Dirac surface states and nature of superconductivity in Noncentrosymmetric BiPd

In non-magnetic bulk materials, inversion symmetry protects the spin degeneracy. If the bulk crystal structure lacks a centre of inversion, however, spin-orbit interactions lift the spin degeneracy, leading to a Rashba metal whose Fermi surfaces exhibit an intricate spin texture. In superconducting Rashba metals a pairing wavefunction constructed from these complex spin structures will generally contain both singlet and triplet character. Here we examine the possible triplet components of the order parameter in noncentrosymmetric BiPd, combining for the first time in a noncentrosymmetric superconductor macroscopic characterization, atomic-scale ultra-low-temperature scanning tunnelling spectroscopy, and relativistic first-principles calculations. While the superconducting state of BiPd appears topologically trivial, consistent with Bardeen-Cooper-Schrieffer theory with an order parameter governed by a single isotropic s-wave gap, we show that the material exhibits Dirac-cone surface states with a helical spin polarization.

Anisotropic superconducting gap and elongated vortices with Caroli-De Gennes-Matricon states in the new superconductor Ta4Pd3Te16

The superconducting state is formed by the condensation of a large number of Cooper pairs. The normal state electronic properties can give significant influence on the superconducting state. For usual type-II superconductors, the vortices are cylinder like with a round cross-section. For many two dimensional superconductors, such as Cuprates, albeit the in-plane anisotropy, the vortices generally have a round shape. In this paper we report results based on the scanning tunnelling microscopy/spectroscopy measurements on a newly discovered superconductor Ta₄Pd₃Te₁₆. The chain-like conducting channels of PdTe₂ in Ta₄Pd₃Te₁₆ make a significant anisotropy of the in-plane Fermi velocity. We suggest at least one anisotropic superconducting gap with gap minima or possible node exists in this multiband system. In addition, elongated vortices are observed with an anisotropy of ξ_||b/ξ_&bottom⊥b ≈ 2.5. Clear Caroli-de Gennes-Matricon states are also observed within the vortex cores. Our results will initiate the study on the elongated vortices and superconducting mechanism in the new superconductor Ta₄Pd₃Te₁₆.

How the vortex lattice of a superconductor becomes disordered: a study by scanning tunneling spectroscopy

Order-disorder transitions take place in many physical systems, but observing them in detail in real materials is difficult. In two- or quasi-two-dimensional systems, the transition has been studied by computer simulations and experimentally in electron sheets, dusty plasmas, colloidal and other systems. Here I show the different stages of defect formation in the vortex lattice of a superconductor while it undergoes an order-disorder transition by presenting real-space images of the lattice from scanning tunneling spectroscopy. When the system evolves from the ordered to the disordered state, the predominant kind of defect changes from dislocation pairs to single dislocations, and finally to defect clusters forming grain boundaries. Correlation functions indicate a hexatic-like state preceding the disordered state. The transition in the microscopic vortex distribution is mirrored by the well-known spectacular second peak effect observed in the macroscopic current density of the superconductor.

Experimental detection of a Majorana mode in the core of a magnetic vortex inside a topological insulator-superconductor Bi(2)Te(3)/NbSe(2) heterostructure

Majorana fermions have been intensively studied in recent years for their importance to both fundamental science and potential applications in topological quantum computing. They are predicted to exist in a vortex core of superconducting topological insulators. However, it is extremely difficult to distinguish them experimentally from other quasiparticle states for the tiny energy difference between Majorana fermions and these states, which is beyond the energy resolution of most available techniques. Here, we circumvent the problem by systematically investigating the spatial profile of the Majorana mode and the bound quasiparticle states within a vortex in Bi(2)Te(3) films grown on a superconductor NbSe(2). While the zero bias peak in local conductance splits right off the vortex center in conventional superconductors, it splits off at a finite distance ∼20  nm away from the vortex center in Bi(2)Te(3). This unusual splitting behavior has never been observed before and could be possibly due to the Majorana fermion zero mode. While the Majorana mode is destroyed by the interaction between vortices, the zero bias peak splits as a conventional superconductor again. This work provides self-consistent evidences of Majorana fermions and also suggests a possible route to manipulating them.

Imaging Josephson vortices on the surface superconductor Si(111)-(√7×√3)-In using a scanning tunneling microscope

We have studied the superconducting Si(111)-(√7×√3)-In surface using a ³He-based low-temperature scanning tunneling microscope. Zero-bias conductance images taken over a large surface area reveal that vortices are trapped at atomic steps after magnetic fields are applied. The crossover behavior from Pearl to Josephson vortices is clearly identified from their elongated shapes along the steps and significant recovery of superconductivity within the cores. Our numerical calculations combined with experiments clarify that these characteristic features are determined by the relative strength of the interterrace Josephson coupling at the atomic step.

Dynamic visualization of nanoscale vortex orbits

Due to the atomic-scale resolution, scanning tunneling microscopy is an ideal technique to observe the smallest objects. Nevertheless, it suffers from very long capturing times in order to investigate dynamic processes at the nanoscale. We address this issue, for vortex matter in NbSe2, by driving the vortices using an ac magnetic field and probing the induced periodic tunnel current modulations. Our results reveal different dynamical modes of the driven vortex lattices. In addition, by recording and synchronizing the time evolution of the tunneling current at each pixel, we visualize the overall dynamics of the vortex lattice with submillisecond time resolution and subnanometer spatial resolution.