Atomic-Scale Andreev Probe of Unconventional Superconductivity

Recent emergence of low-dimensional unconventional superconductors and their exotic interface properties calls for new approaches to probe the pairing symmetry, a fundamental and frequently elusive property of the superconducting condensate. Here, we introduce the unique capability of tunneling Andreev reflection (TAR) to probe unconventional pairing symmetry, utilizing the sensitivity of this technique to specific Andreev reflections. Specifically, suppression of the lowest-order Andreev reflection due to quantum interference but emergence of the higher-order Andreev processes provides direct evidence of the sign-changing order parameter in the paradigmatic FeSe superconductor. TAR spectroscopy also reveals two superconducting gaps, points to a possibility of a nodal gap structure, and directly confirms that superconductivity is locally suppressed along the nematic twin boundary, with preferential and near-complete suppression of the larger energy gap. Our findings therefore enable new, atomic-scale insight into microscopic, inhomogeneous, and interfacial properties of emerging quantum materials.

Pair density wave state in a monolayer high-Tc iron-based superconductor

The pair density wave (PDW) is an extraordinary superconducting state in which Cooper pairs carry non-zero momentum^1,2. Evidence for the existence of intrinsic PDW order in high-temperature (high-T_c) cuprate superconductors^3,4 and kagome superconductors⁵ has emerged recently. However, the PDW order in iron-based high-T_c superconductors has not been observed experimentally. Here, using scanning tunnelling microscopy and spectroscopy, we report the discovery of the PDW state in monolayer iron-based high-T_c Fe(Te,Se) films grown on SrTiO₃(001) substrates. The PDW state with a period of λ ≈ 3.6a_Fe (a_Fe is the distance between neighbouring Fe atoms) is observed at the domain walls by the spatial electronic modulations of the local density of states, the superconducting gap and the π-phase shift boundaries of the PDW around the vortices of the intertwined charge density wave order. The discovery of the PDW state in the monolayer Fe(Te,Se) film provides a low-dimensional platform to study the interplay between the correlated electronic states and unconventional Cooper pairing in high-T_c superconductors.

Nematically Templated Vortex Lattices in Superconducting FeSe

New pathways to controlling the morphology of superconducting vortex lattices─and their subsequent dynamics─are required to guide and scale vortex world-lines into a computing platform. We have found that the nematic twin boundaries align superconducting vortices in the adjacent terraces due to the incommensurate potential between vortices surrounding twin boundaries and those trapped within them. With the varying density and morphology of twin boundaries, the vortex lattice assumes several distinct structural phases, including square, regular, and irregular one-dimensional lattices. Through concomitant analysis of vortex lattice models, we have inferred the characteristic energetics of the twin boundary potential and furthermore predicted the existence of geometric size effects as a function of increasing confinement by the twin boundaries. These findings extend the ideas of directed control over vortex lattices to intrinsic topological defects and their self-organized networks, which have direct implications for the future design and control of strain-based topological quantum computing architectures.

Inhomogeneous Superconductivity Onset in FeSe Studied by Transport Properties

Heterogeneous superconductivity onset is a common phenomenon in high-Tc superconductors of both the cuprate and iron-based families. It is manifested by a fairly wide transition from the metallic to zero-resistance states. Usually, in these strongly anisotropic materials, superconductivity (SC) first appears as isolated domains. This leads to anisotropic excess conductivity above Tc, and the transport measurements provide valuable information about the SC domain structure deep within the sample. In bulk samples, this anisotropic SC onset gives an approximate average shape of SC grains, while in thin samples, it also indicates the average size of SC grains. In this work, both interlayer and intralayer resistivity were measured as a function of temperature in FeSe samples of various thicknesses. To measure the interlayer resistivity, FeSe mesa structures oriented across the layers were fabricated using FIB. As the sample thickness decreases, a significant increase in superconducting transition temperature Tc is observed: Tc raises from 8 K in bulk material to 12 K in microbridges of thickness ∼40 nm. We applied analytical and numerical calculations to analyze these and earlier data and find the aspect ratio and size of the SC domains in FeSe consistent with our resistivity and diamagnetic response measurements. We propose a simple and fairly accurate method for estimating the aspect ratio of SC domains from Tc anisotropy in samples of various small thicknesses. The relationship between nematic and superconducting domains in FeSe is discussed. We also generalize the analytical formulas for conductivity in heterogeneous anisotropic superconductors to the case of elongated SC domains of two perpendicular orientations with equal volume fractions, corresponding to the nematic domain structure in various Fe-based superconductors.

Observations of Nematicity, Dopants, and Zero-Bias Conductance Peaks for the Ca0.9La0.1FeAs2 Superconductor

Ca_1-xLa_xFeAs₂ (CLFA112) belongs to a new family of Fe-based superconductors (FeSCs) and has a unique crystal structure featuring an arsenic zigzag chain layer, which has been proposed to be a possible two-dimensional topological insulator. This suggests that CLFA112 is a potential topological superconductor-a platform to realize Majorana fermions. Up to now, even a clear superconducting (SC) gap in CLFA112 has never been observed, and the SC properties of CLFA112 remain largely elusive. In this letter, we report the results of an atomic-scale investigation of the electronic structure of CLFA112 crystals using low-temperature scanning tunneling microscopy (STM). We revealed four different types of surfaces exhibiting distinct electronic properties, with all surfaces displaying dominating 2 × 1 surface reconstructions. On a Ca/La layer on top of an FeAs layer, a clear SC gap of ~12 mV was observed only at the crevices (vacancies) where the FeAs layer can be directly accessed. Remarkably, the FeAs termination layer displayed a dispersing nematic modulation both in real and q space. We also present peculiar zero-bias conductance peaks for the very As chain layer that is believed to exhibit a topological edge state as well as the influence of La dopants on the As chain layer.

Anomalous Contribution to the Nematic Electronic States from the Structural Transition in FeSe Revealed by Time- and Angle-Resolved Photoemission Spectroscopy

High-resolution time- and angle-resolved photoemission measurements were made on FeSe superconductors. With ultrafast photoexcitation, two critical excitation fluences that correspond to two ultrafast electronic phase transitions were found only in the d_{yz}-orbit-derived band near the Brillouin-zone center within our time and energy resolution. Upon comparison to the detailed temperature dependent measurements, we conclude that there are two equilibrium electronic phase transitions (at approximately 90 and 120 K) above the superconducting transition temperature, and an anomalous contribution on the scale of 10 meV to the nematic states from the structural transition is experimentally determined. Our observations strongly suggest that the electronic phase transition at 120 K must be taken into account in the energy band development of FeSe, and, furthermore, the contribution of the structural transition plays an important role in the nematic phase of iron-based high-temperature superconductors.

Dirac-cone-like electronic states on nematic antiferromagnetic FeSe and FeTe

We investigate the Dirac-cone-like (DCL) topological electronic properties of nematic-like antiferromagnetic (AFM) states of monolayer FeSe and FeTe designed artificially through first-principles calculations and Wannier-function-based tight-binding (WFTB) method. Our calculations reveal most of them have a pair of DCL bands on the Γ-Xline in the Brillouin zone (BZ) near the Fermi level and open a gap of about 20 meV in the absence and presence of spin-orbit coupling (SOC), respectively, similar to the lowest-energy pair-checkerboard AFM FeSe. We further confirm that they are weak topological insulators based on nonzeroZ₂and fragile surface states, which are calculated by the WFTB method. For FeSe and FeTe in pair-checkerboard AFM states, we find that the in-plane compression strain in a certain range can give rise to another pair of DCL bands located on the Γ-X' line in the BZ. In addition, the magnetic moments, energies, and Fe-Se/Te distances for various nematic-like AFM configurations are presented. These calculations the combining effect of magnetism and topology in a single material and the understanding of the superconducting phenomena in iron-based FeSe and FeTe.

Iron pnictides and chalcogenides: a new paradigm for superconductivity

Superconductivity is a remarkably widespread phenomenon that is observed in most metals cooled to very low temperatures. The ubiquity of such conventional superconductors, and the wide range of associated critical temperatures, is readily understood in terms of the well-known Bardeen-Cooper-Schrieffer theory. Occasionally, however, unconventional superconductors are found, such as the iron-based materials, which extend and defy this understanding in unexpected ways. In the case of the iron-based superconductors, this includes the different ways in which the presence of multiple atomic orbitals can manifest in unconventional superconductivity, giving rise to a rich landscape of gap structures that share the same dominant pairing mechanism. In addition, these materials have also led to insights into the unusual metallic state governed by the Hund's interaction, the control and mechanisms of electronic nematicity, the impact of magnetic fluctuations and quantum criticality, and the importance of topology in correlated states. Over the fourteen years since their discovery, iron-based superconductors have proven to be a testing ground for the development of novel experimental tools and theoretical approaches, both of which have extensively influenced the wider field of quantum materials.

Observation of an electronic order along [110] direction in FeSe

Multiple ordered states have been observed in unconventional superconductors. Here, we apply scanning tunneling microscopy to probe the intrinsic ordered states in FeSe, the structurally simplest iron-based superconductor. Besides the well-known nematic order along [100] direction, we observe a checkerboard charge order in the iron lattice, which we name a [110] electronic order in FeSe. The [110] electronic order is robust at 77 K, accompanied with the rather weak [100] nematic order. At 4.5 K, The [100] nematic order is enhanced, while the [110] electronic order forms domains with reduced correlation length. In addition, the collective [110] order is gaped around [−40, 40] meV at 4.5 K. The observation of this exotic electronic order may shed new light on the origin of the ordered states in FeSe.

Understanding the relation of different electronic orders in high temperature superconductors is of fundamental interest. Here, the authors observe a checkerboard charge order along [110] direction of FeSe.

Nanoscale decoupling of electronic nematicity and structural anisotropy in FeSe thin films

In a material prone to a nematic instability, anisotropic strain in principle provides a preferred symmetry-breaking direction for the electronic nematic state to follow. This is consistent with experimental observations, where electronic nematicity and structural anisotropy typically appear hand-in-hand. In this work, we discover that electronic nematicity can be locally decoupled from the underlying structural anisotropy in strain-engineered iron-selenide (FeSe) thin films. We use heteroepitaxial molecular beam epitaxy to grow FeSe with a nanoscale network of modulations that give rise to spatially varying strain. We map local anisotropic strain by analyzing scanning tunneling microscopy topographs, and visualize electronic nematic domains from concomitant spectroscopic maps. While the domains form so that the energy of nemato-elastic coupling is minimized, we observe distinct regions where electronic nematic ordering fails to flip direction, even though the underlying structural anisotropy is locally reversed. The findings point towards a nanometer-scale stiffness of the nematic order parameter.

Electronic Structures and Surface Reconstructions in Magnetic Superconductor RbEuFe4As4

In pnictide RbEuFe₄As₄, superconductivity sets in at 36 K and coexists, below 15-19 K, with the long-range magnetic ordering of Eu 4f spins. Here we report scanning tunneling experiments performed on cold-cleaved single crystals of the compound. The data revealed the coexistence of large Rb-terminated and small Eu-terminated terraces, both manifesting 1 × 2 and 2×2 reconstructions. On 2×2 surfaces, a hidden electronic order with a period ∼5 nm was discovered. A superconducting gap of ∼7 meV was seen to be strongly filled with quasiparticle states. The tunneling spectra compared with density functional theory calculations confirmed that flat electronic bands due to Eu 4f orbitals are situated ∼1.8 eV below the Fermi level and thus do not contribute directly to Cooper pair formation.

Electronic nematicity in Sr2RuO4

We have measured the angle-resolved transverse resistivity (ARTR), a sensitive indicator of electronic anisotropy, in high-quality thin films of the unconventional superconductor Sr₂RuO₄ grown on various substrates. The ARTR signal, heralding the electronic nematicity or a large nematic susceptibility, is present and substantial already at room temperature and grows by an order of magnitude upon cooling down to 4 K. In Sr₂RuO₄ films deposited on tetragonal substrates the highest-conductivity direction does not coincide with any crystallographic axis. In films deposited on orthorhombic substrates it tends to align with the shorter axis; however, the magnitude of the anisotropy stays the same despite the large lattice distortion. These are strong indications of actual or incipient electronic nematicity in Sr₂RuO₄.

Charge ordering in high-temperature superconductors visualized by scanning tunneling microscopy

Since the discovery of stripe order in La_1.6-x Nd_0.4Sr _x CuO₄ superconductors in 1995, charge ordering in cuprate superconductors has been intensively studied by various experimental techniques. Among these studies, scanning tunneling microscope (STM) plays an irreplaceable role in determining the real space structures of charge ordering. STM imaging of different families of cuprates over a wide range of doping levels reveal similar checkerboard-like patterns, indicating that such a charge ordered state is likely a ubiquitous and intrinsic characteristic of cuprate superconductors, which may shed light on understanding the mechanism of high-temperature superconductivity. In another class of high-temperature superconductors, iron-based superconductors, STM studies reveal several charge ordered states as well, but their real-space patterns and the interplay with superconductivity are markedly different among different materials. In this paper, we present a brief review on STM studies of charge ordering in these two classes of high-temperature superconductors. Possible origins of charge ordering and its interplay with superconductivity will be discussed.

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.

Cooper Pair Density of Bi2Sr2CaCu2O8+ x in Atomic scale at 4.2 K

In pursuit of the elusive mechanism of high- T _C superconductors (HTSC), spectroscopic imaging scanning tunneling microscopy (SI-STM) is an indispensable tool for surveying local properties of HTSC. Since a conventional STM utilizes metal tips, which allow the examination of only quasiparticles and not superconducting (SC) pairs, Josephson tunneling using STM has been demonstrated by many authors in the past. An atomically resolved scanning Josephson tunneling microscopy (SJTM), however, was realized only recently on Bi₂Sr₂CaCu₂O_{8+ x} (Bi-2212) below 50 mK and on the Pb(110) surface at 20 mK. Here we report the atomically resolved SJTM on Bi₂Sr₂CaCu₂O_{8+ x} at 4.2 K using Bi-2212 tips created in situ. The I- V characteristics show clear zero bias conductance peaks following Ambegaokar-Baratoff (AB) theory. A gap map was produced for the first time using an atomically resolved Josephson critical current map I _C( r) and AB theory. Surprisingly, we found that this new gap map is anticorrelated to the gap map produced by a conventional method relying on the coherence peaks. Quasiparticle resonance due to a single isolated zinc atom impurity was also observed by SJTM, indicating that atomically resolved SJTM was achieved at 4.2 K. Our result provides a starting point for realizing SJTM at even higher temperatures, rendering possible investigation of the existence of SC pairs in HTSC above the T _C.

Anomalous transverse resistance in 122-type iron-based superconductors

The study of transverse resistance of superconductors is essential to understand the transition to superconductivity. Here, we investigated the in-plane transverse resistance of Ba_0.5K_0.5Fe₂As₂ superconductors, based on ultra-thin micro-bridges fabricated from optimally doped single crystals. An anomalous transverse resistance was found at temperatures around the superconducting transition, although magnetic order or structure distortion are absent in the optimal doping case. With the substitution of magnetic and nonmagnetic impurities into the superconducting layer, the anomalous transverse resistance phenomenon is dramatically enhanced. We find that anisotropic scattering or the superconducting electronic nematic state related with the superconducting transition may contribute to this phenomenon.

Edge States at Nematic Domain Walls in FeSe Films

Quantum spin Hall (QSH) effect is an intriguing phenomenon arising from the helical edge states in two-dimensional topological insulators. We use molecular beam epitaxy (MBE) to prepare FeSe films with atomically sharp nematic domain boundaries, where tensile strains, nematicity suppression, and topological band inversion are simultaneously achieved. Using scanning tunneling microscopy (STM), we observe edge states at the Fermi level that spatially distribute as two distinct strips in the vicinity of the domain boundaries. At the end point of the boundaries, a bound state at the Fermi level is further observed. The topological origin of the edge states is supported by density functional theory calculations. Our findings not only demonstrate a candidate for QSH states but also provide a new pathway to realize topological superconductivity in a single-component film.

Multi-band effects in in-plane resistivity anisotropy of strain-detwinned disordered Ba(Fe1-xRux)2As2

In-plane resistivity anisotropy was measured in strain-detwinned as-grown and partially annealed samples of isovalently-substituted [Formula: see text] ([Formula: see text]) and the results were contrasted with previous reports on anneal samples with low residual resistivity. In samples with high residual resistivity, detwinned with application of strain, the difference of the two components of in-plane resistivity in the orthorhombic phase, [Formula: see text], was found to obey Matthiessen rule irrespective of sample composition, which is in stark contrast with observations on annealed samples. Our findings are consistent with two-band transport model in which contribution from high mobility carriers of small pockets of the Fermi surface has negligible anisotropy of residual resistivity and is eliminated by disorder. Our finding suggests that magnetic/nematic order has dramatically different effect on different parts of the Fermi surface. It predominantly affects inelastic scattering for small pocket high mobility carriers and elastic impurity scattering for larger sheets of the Fermi surface.

Surface reconstruction and charge modulation in BaFe2As2 superconducting film

Whether or not epitaxially grown superconducting films have the same bulk-like superconducting properties is an important concern. We report the structure and the electronic properties of epitaxially grown Ba(Fe_1-x Co _x )₂As₂ films using scanning tunneling microscopy and scanning tunneling spectroscopy (STS). This film showed a different surface structure, [Formula: see text]R45° reconstruction, from those of as-cleaved surfaces from bulk crystals. The electronic structure of the grown film is different from that in bulk, and it is notable that the film exhibits the same superconducting transport properties. We found that the superconducting gap at the surface is screened at the Ba layer surface in STS measurements, and the charge density wave was observed at the surface in sample in the superconducting state.

Discovery of a strain-stabilised smectic electronic order in LiFeAs

In many high temperature superconductors, small orthorhombic distortions of the lattice structure result in surprisingly large symmetry breaking of the electronic states and macroscopic properties, an effect often referred to as nematicity. To directly study the impact of symmetry-breaking lattice distortions on the electronic states, using low-temperature scanning tunnelling microscopy we image at the atomic scale the influence of strain-tuned lattice distortions on the correlated electronic states in the iron-based superconductor LiFeAs, a material which in its ground state is tetragonal with four-fold (C4) symmetry. Our experiments uncover a new strain-stabilised modulated phase which exhibits a smectic order in LiFeAs, an electronic state which not only breaks rotational symmetry but also reduces translational symmetry. We follow the evolution of the superconducting gap from the unstrained material with C4 symmetry through the new smectic phase with two-fold (C2) symmetry and charge-density wave order to a state where superconductivity is completely suppressed.

Nematicity, magnetism and superconductivity in FeSe

Iron-based superconductors are well known for their complex interplay between structure, magnetism and superconductivity. FeSe offers a particularly fascinating example. This material has been intensely discussed because of its extended nematic phase, whose relationship with magnetism is not obvious. Superconductivity in FeSe is highly tunable, with the superconducting transition temperature, T _c, ranging from 8 K in bulk single crystals at ambient pressure to almost 40 K under pressure or in intercalated systems, and to even higher temperatures in thin films. In this topical review, we present an overview of nematicity, magnetism and superconductivity, and discuss the interplay of these phases in FeSe. We focus on bulk FeSe and the effects of physical pressure and chemical substitutions as tuning parameters. The experimental results are discussed in the context of the well-studied iron-pnictide superconductors and interpretations from theoretical approaches are presented.

Transverse fields to tune an Ising-nematic quantum phase transition

The paradigmatic example of a continuous quantum phase transition is the transverse field Ising ferromagnet. In contrast to classical critical systems, whose properties depend only on symmetry and the dimension of space, the nature of a quantum phase transition also depends on the dynamics. In the transverse field Ising model, the order parameter is not conserved, and increasing the transverse field enhances quantum fluctuations until they become strong enough to restore the symmetry of the ground state. Ising pseudospins can represent the order parameter of any system with a twofold degenerate broken-symmetry phase, including electronic nematic order associated with spontaneous point-group symmetry breaking. Here, we show for the representative example of orbital-nematic ordering of a non-Kramers doublet that an orthogonal strain or a perpendicular magnetic field plays the role of the transverse field, thereby providing a practical route for tuning appropriate materials to a quantum critical point. While the transverse fields are conjugate to seemingly unrelated order parameters, their nontrivial commutation relations with the nematic order parameter, which can be represented by a Berry-phase term in an effective field theory, intrinsically intertwine the different order parameters.

Nematic superconducting state in iron pnictide superconductors

Nematic order often breaks the tetragonal symmetry of iron-based superconductors. It arises from regular structural transition or electronic instability in the normal phase. Here, we report the observation of a nematic superconducting state, by measuring the angular dependence of the in-plane and out-of-plane magnetoresistivity of Ba_0.5K_0.5Fe₂As₂ single crystals. We find large twofold oscillations in the vicinity of the superconducting transition, when the direction of applied magnetic field is rotated within the basal plane. To avoid the influences from sample geometry or current flow direction, the sample was designed as Corbino-shape for in-plane and mesa-shape for out-of-plane measurements. Theoretical analysis shows that the nematic superconductivity arises from the weak mixture of the quasi-degenerate s-wave and d-wave components of the superconducting condensate, most probably induced by a weak anisotropy of stresses inherent to single crystals.

Nematic electronic order is rare and its onset often indicates a phase transition. Here, Li et al. report a nematic superconducting state in Ba_0.5K_0.5Fe₂As₂ by measuring the angular dependence of the in-plane and out-of-plane magnetoresistivity.

Spin-wave excitations in the spin-density wave state of doped iron pnictides

We investigate spin-wave excitations in the spin-density wave state of doped iron pnictides within a five-orbital model. We find that the excitations along ([Formula: see text])  →  ([Formula: see text]) are very sensitive to dopings whereas they do not exhibit a similar sensitivity along ([Formula: see text])  →  ([Formula: see text]). Secondly, the ellipticity of the elliptical ring-like excitations around ([Formula: see text]) is also very much dependent on doping. Thirdly, the spin-wave spectral weight shifts towards the low-energy region as it moves away from zero doping. We find several features to be in qualitative agreement with the inelastic neutron-scattering measurements for the doped pnictides.

Low-energy microscopic models for iron-based superconductors: a review

The development of sensible microscopic models is essential to elucidate the normal-state and superconducting properties of the iron-based superconductors. Because these materials are mostly metallic, a good starting point is an effective low-energy model that captures the electronic states near the Fermi level and their interactions. However, in contrast to cuprates, iron-based high-T _c compounds are multi-orbital systems with Hubbard and Hund interactions, resulting in a rather involved 10-orbital lattice model. Here we review different minimal models that have been proposed to unveil the universal features of these systems. We first review minimal models defined solely in the orbital basis, which focus on a particular subspace of orbitals, or solely in the band basis, which rely only on the geometry of the Fermi surface. The former, while providing important qualitative insight into the role of the orbital degrees of freedom, do not distinguish between high-energy and low-energy sectors and, for this reason, generally do not go beyond mean-field. The latter allow one to go beyond mean-field and investigate the interplay between superconducting and magnetic orders as well as Ising-nematic order. However, they cannot capture orbital-dependent features like spontaneous orbital order. We then review recent proposals for a minimal model that operates in the band basis but fully incorporates the orbital composition and symmetries of the low-energy excitations. We discuss the results of the renormalization group study of such a model, particularly of the interplay between superconductivity, magnetism, and spontaneous orbital order, and compare theoretical predictions with experiments on iron pnictides and chalcogenides. We also discuss the impact of the glide-plane symmetry on the low-energy models, highlighting the key role played by the spin-orbit coupling.

Nanoscale electrodynamics of strongly correlated quantum materials

Electronic, magnetic, and structural phase inhomogeneities are ubiquitous in strongly correlated quantum materials. The characteristic length scales of the phase inhomogeneities can range from atomic to mesoscopic, depending on their microscopic origins as well as various sample dependent factors. Therefore, progress with the understanding of correlated phenomena critically depends on the experimental techniques suitable to provide appropriate spatial resolution. This requirement is difficult to meet for some of the most informative methods in condensed matter physics, including infrared and optical spectroscopy. Yet, recent developments in near-field optics and imaging enabled a detailed characterization of the electromagnetic response with a spatial resolution down to 10 nm. Thus it is now feasible to exploit at the nanoscale well-established capabilities of optical methods for characterization of electronic processes and lattice dynamics in diverse classes of correlated quantum systems. This review offers a concise description of the state-of-the-art near-field techniques applied to prototypical correlated quantum materials. We also discuss complementary microscopic and spectroscopic methods which reveal important mesoscopic dynamics of quantum materials at different energy scales.

Deep data mining in a real space: separation of intertwined electronic responses in a lightly doped BaFe2As2

Electronic interactions present in material compositions close to the superconducting dome play a key role in the manifestation of high-T _c superconductivity. In many correlated electron systems, however, the parent or underdoped states exhibit strongly inhomogeneous electronic landscape at the nanoscale that may be associated with competing, coexisting, or intertwined chemical disorder, strain, magnetic, and structural order parameters. Here we demonstrate an approach based on a combination of scanning tunneling microscopy/spectroscopy and advanced statistical learning for an automatic separation and extraction of statistically significant electronic behaviors in the spin density wave regime of a lightly (∼1%) gold-doped BaFe₂As₂. We show that the decomposed STS spectral features have a direct relevance to fundamental physical properties of the system, such as SDW-induced gap, pseudogap-like state, and impurity resonance states.

Direct observation of electronic-liquid-crystal phase transitions and their microscopic origin in La1/3Ca2/3MnO3

The ground-state electronic order in doped manganites is frequently associated with a lattice modulation, contributing to their many interesting properties. However, measuring the thermal evolution of the lattice superstructure with reciprocal-space probes alone can lead to ambiguous results with competing interpretations. Here we provide direct observations of the evolution of the superstructure in La1/3Ca2/3MnO3 in real space, as well as reciprocal space, using transmission electron microscopic (TEM) techniques. We show that the transitions are the consequence of a proliferation of dislocations plus electronic phase separation. The resulting states are well described by the symmetries associated with electronic-liquid-crystal (ELC) phases. Moreover, our results resolve the long-standing controversy over the origin of the incommensurate superstructure and suggest a new structural model that is consistent with recent theoretical calculations.

Dispersing artifacts in FT-STS: a comparison of set point effects across acquisition modes

Fourier-transform scanning tunnelling spectroscopy (FT-STS), or quasiparticle interference, has become an influential tool for the study of a wide range of important materials in condensed matter physics. However, FT-STS in complex materials is often challenging to interpret, requiring significant theoretical input in many cases, making it crucial to understand potential artifacts of the measurement. Here, we compare the most common modes of acquiring FT-STS data and show through both experiment and simulations that artifact features can arise that depend on how the tip height is stabilized throughout the course of the measurement. The most dramatic effect occurs when a series of dI/dV maps at different energies are acquired with simultaneous constant current feedback; here a feature that disperses in energy appears that is not observed in other measurement modes. Such artifact features are similar to those arising from real physical processes in the sample and are susceptible to misinterpretation.

Using Nonequilibrium Dynamics to Probe Competing Orders in a Mott-Peierls System

Competition between ordered phases, and their associated phase transitions, are significant in the study of strongly correlated systems. Here, we examine one aspect, the nonequilibrium dynamics of a photoexcited Mott-Peierls system, using an effective Peierls-Hubbard model and exact diagonalization. Near a transition where spin and charge become strongly intertwined, we observe antiphase dynamics and a coupling-strength-dependent suppression or enhancement in the static structure factors. The renormalized bosonic excitations coupled to a particular photoexcited electron can be extracted, which provides an approach for characterizing the underlying bosonic modes. The results from this analysis for different electronic momenta show an uneven softening due to a stronger coupling near k_{F}. This behavior reflects the strong link between the fermionic momenta, the coupling vertices, and ultimately, the bosonic susceptibilities when multiple phases compete for the ground state of the system.

Strong interplay between stripe spin fluctuations, nematicity and superconductivity in FeSe

In iron-based superconductors the interactions driving the nematic order (that breaks four-fold rotational symmetry in the iron plane) may also mediate the Cooper pairing. The experimental determination of these interactions, which are believed to depend on the orbital or the spin degrees of freedom, is challenging because nematic order occurs at, or slightly above, the ordering temperature of a stripe magnetic phase. Here, we study FeSe (ref. )-which exhibits a nematic (orthorhombic) phase transition at Ts = 90 K without antiferromagnetic ordering-by neutron scattering, finding substantial stripe spin fluctuations coupled with the nematicity that are enhanced abruptly on cooling through Ts. A sharp spin resonance develops in the superconducting state, whose energy (∼4 meV) is consistent with an electron-boson coupling mode revealed by scanning tunnelling spectroscopy. The magnetic spectral weight in FeSe is found to be comparable to that of the iron arsenides. Our results support recent theoretical proposals that both nematicity and superconductivity are driven by spin fluctuations.

Liquid crystal phases of two-dimensional dipolar gases and Berezinskii-Kosterlitz-Thouless melting

Liquid crystals are phases of matter intermediate between crystals and liquids. Whereas classical liquid crystals have been known for a long time and are used in electro-optical displays, much less is known about their quantum counterparts. There is growing evidence that quantum liquid crystals play a central role in many electron systems including high temperature superconductors, but a quantitative understanding is lacking due to disorder and other complications. Here, we analyse the quantum phase diagram of a two-dimensional dipolar gas, which exhibits stripe, nematic and supersolid phases. We calculate the stiffness constants determining the stability of the nematic and stripe phases, and the melting of the stripes set by the proliferation of topological defects is analysed microscopically. Our results for the critical temperatures of these phases demonstrate that a controlled study of the interplay between quantum liquid and superfluid phases is within experimental reach for the first time, using dipolar gases.

Nematic Resonance in the Raman Response of Iron-Based Superconductors

In a fully gapped superconductor the electronic Raman response has a pair-breaking peak at twice the superconducting gap Δ, if the Bogoliubov excitations are uncorrelated. Motivated by the iron based superconductors, we study how this peak is modified if the superconducting phase hosts a nematic-structural quantum critical point. We show that, upon approaching this point by tuning, e.g., doping, the growth of nematic correlations between the quasiparticles transforms the pair-breaking peak into a nematic resonance. The mode energy is below 2Δ, and stays finite at the quantum critical point, where its spectral weight is sharply enhanced. The latter is consistent with recent experiments on electron-doped iron based superconductors and provides direct evidence of nematic correlations in their superconducting phases.

Evidence for orbital order and its relation to superconductivity in FeSe0.4Te0.6

The emergence of nematic electronic states accompanied by a structural phase transition is a recurring theme in many correlated electron materials, including the high-temperature copper oxide- and iron-based superconductors. We provide evidence for nematic electronic states in the iron-chalcogenide superconductor FeSe0.4Te0.6 from quasi-particle scattering detected in spectroscopic maps. The symmetry-breaking states persist above T c into the normal state. We interpret the scattering patterns by comparison with quasi-particle interference patterns obtained from a tight-binding model, accounting for orbital ordering. The relation to superconductivity and the influence on the coherence length are discussed.

Origin of the Resistive Anisotropy in the Electronic Nematic Phase of BaFe(2)As(2) Revealed by Optical Spectroscopy

We perform, as a function of uniaxial stress, an optical-reflectivity investigation of the representative "parent" ferropnictide BaFe(2)As(2) in a broad spectral range, across the tetragonal-to-orthorhombic phase transition and the onset of the long-range antiferromagnetic (AFM) order. The infrared response reveals that the dc transport anisotropy in the orthorhombic AFM state is determined by the interplay between the Drude spectral weight and the scattering rate, but that the dominant effect is clearly associated with the metallic spectral weight. In the paramagnetic tetragonal phase, though, the dc resistivity anisotropy of strained samples is almost exclusively due to stress-induced changes in the Drude weight rather than in the scattering rate, definitively establishing the anisotropy of the Fermi surface parameters as the primary effect driving the dc transport properties in the electronic nematic state.

ARPES measurements of the superconducting gap of Fe-based superconductors and their implications to the pairing mechanism

Its direct momentum sensitivity confers to angle-resolved photoemission spectroscopy (ARPES) a unique perspective in investigating the superconducting gap of multi-band systems. In this review we discuss ARPES studies on the superconducting gap of high-temperature Fe-based superconductors. We show that while Fermi-surface-driven pairing mechanisms fail to provide a universal scheme for the Fe-based superconductors, theoretical approaches based on short-range interactions lead to a more robust and universal description of superconductivity in these materials. Our findings are also discussed in the broader context of unconventional superconductivity.

Orbital nematic order and interplay with magnetism in the two-orbital Hubbard model

Motivated by the recent angle-resolved photoemission spectroscopy (ARPES) on FeSe and iron pnictide families of iron-based superconductors, we have studied the orbital nematic order and its interplay with antiferromagnetism within the two-orbital Hubbard model. We used random phase approximation (RPA) to calculate the dependence of the orbital and magnetic susceptibilities on the strength of interactions and electron density (doping). To account for strong electron correlations not captured by RPA, we further employed non-perturbative variational cluster approximation (VCA) capable of capturing symmetry broken magnetic and orbitally ordered phases. Both approaches show that the electron and hole doping affect the two orders differently. While hole doping tends to suppress both magnetism and orbital ordering, the electron doping suppresses magnetism faster. Crucially, we find a realistic parameter regime for moderate electron doping that stabilizes orbital nematicity in the absence of long-range antiferromagnetic order. This is reminiscent of the non-magnetic orbital nematic phase observed recently in FeSe and a number of iron pnictide materials and raises the possibility that at least in some cases, the observed electronic nematicity may be primarily due to orbital rather than magnetic fluctuations.

Spin-orbit interactions and the nematicity observed in the fe-based superconductors

High-resolution angle-resolved photoelectron spectroscopy is used to examine the electronic band structure of FeTe_{0.5}Se_{0.5} near the Brillouin zone center. A consistent separation of the α_{1} and α_{2} bands is observed with little k_{z} dependence of the α_{1} band. First-principles calculations for bulk and thin films demonstrate that the antiferromagnetic coupling between the Fe atoms and hybridization-induced spin-orbit effects lifts the degeneracy of the Fe d_{xz} and d_{yz} orbitals at the zone center leading to orbital ordering. These experimental and computational results provide a natural microscopic basis for the nematicity observed in the Fe-based superconductors.

The form and origin of orbital ordering in the electronic nematic phase of iron-based superconductors

We investigated the form of orbital ordering in the electronic nematic phase of iron-based superconductors by applying a group theoretical analysis on a realistic five-band model. We find the orbital order can be either of the inter-orbital s-wave form or intra-orbital d-waveform. From the comparison with existing ARPES measurements of band splitting, we find the orbital ordering in the 122 system is dominated by an intra-orbital d-wave component, while that of the 111 system is dominated by an inter-orbital s-wave component. We find both forms of orbital order are strongly entangled with the nematicity in the spin correlation of the system.The condensation energy of the magnetic ordered phase is found to be significantly improved (by more than 20%) when the degeneracy between the (π, 0) and (0, π) ordering pattern is lifted by the orbital order. We argue there should be a large difference in both the scattering rate and the size of the possible pseudogap on the electron pocket around the X = (π, 0) and Y = (0, π) point in the electronic nematic phase. We propose this as a possible origin for the observed nematicity in resistivity measurements.

Effects of Lifshitz transition on charge transport in magnetic phases of Fe-based superconductors

The unusual temperature dependence of the resistivity and its in-plane anisotropy observed in the Fe-based superconducting materials, particularly Ba(Fe_{1-x}Co_{x})_{2}As_{2}, has been a long-standing puzzle. Here, we consider the effect of impurity scattering on the temperature dependence of the average resistivity within a simple two-band model of a dirty spin density wave metal. The sharp drop in resistivity below the Néel temperature T_{N} in the parent compound can only be understood in terms of a Lifshitz transition following Fermi surface reconstruction upon magnetic ordering. We show that the observed resistivity anisotropy in this phase, arising from nematic defect structures, is affected by the Lifshitz transition as well.

Neutron-scattering measurements of spin excitations in LaFeAsO and Ba(Fe(0.953)Co(0.047))(2)As(2): evidence for a sharp enhancement of spin fluctuations by nematic order

Inelastic neutron scattering is employed to investigate the impact of electronic nematic order on the magnetic spectra of LaFeAsO and Ba(Fe(0.953)Co(0.047))(2)As(2). These materials are ideal to study the paramagnetic-nematic state, since the nematic order, signaled by the tetragonal-to-orthorhombic transition at T(S), sets in well above the stripe antiferromagnetic ordering at T(N). We find that the temperature-dependent dynamic susceptibility displays an anomaly at T(S) followed by a sharp enhancement in the spin-spin correlation length, revealing a strong feedback effect of nematic order on the low-energy magnetic spectrum. Our findings can be consistently described by a model that attributes the structural or nematic transition to magnetic fluctuations, and unveils the key role played by nematic order in promoting the long-range stripe antiferromagnetic order in iron pnictides.

Intra-unit-cell nematic charge order in the titanium-oxypnictide family of superconductors

Understanding the role played by broken-symmetry states such as charge, spin and orbital orders in the mechanism of emergent properties, such as high-temperature superconductivity, is a major current topic in materials research. That the order may be within one unit cell, such as nematic, was only recently considered theoretically, but its observation in the iron-pnictide and doped cuprate superconductors places it at the forefront of current research. Here, we show that the recently discovered BaTi2Sb2O superconductor and its parent compound BaTi2As2O form a symmetry-breaking nematic ground state that can be naturally explained as an intra-unit-cell nematic charge order with d-wave symmetry, pointing to the ubiquity of the phenomenon. These findings, together with the key structural features in these materials being intermediate between the cuprate and iron-pnictide high-temperature superconducting materials, render the titanium oxypnictides an important new material system to understand the nature of nematic order and its relationship to superconductivity.

Reconstruction of band structure induced by electronic nematicity in an FeSe superconductor

We have performed high-resolution angle-resolved photoemission spectroscopy on an FeSe superconductor (T_{c}∼8  K), which exhibits a tetragonal-to-orthorhombic structural transition at T_{s}∼90  K. At low temperature, we found splitting of the energy bands as large as 50 meV at the M point in the Brillouin zone, likely caused by the formation of electronically driven nematic states. This band splitting persists up to T∼110  K, slightly above T_{s}, suggesting that the structural transition is triggered by the electronic nematicity. We have also revealed that at low temperature the band splitting gives rise to a van Hove singularity within 5 meV of the Fermi energy. The present result strongly suggests that this unusual electronic state is responsible for the unconventional superconductivity in FeSe.

Emergent defect states as a source of resistivity anisotropy in the nematic phase of iron pnictides

We consider the role of potential scatterers in the nematic phase of Fe-based superconductors above the transition temperature to the (π, 0) magnetic state but below the orthorhombic structural transition. The anisotropic spin fluctuations in this region can be frozen by disorder, to create elongated magnetic droplets whose anisotropy grows as the magnetic transition is approached. Such states act as strong anisotropic defect potentials that scatter with much higher probability perpendicular to their length than parallel, although the actual crystal symmetry breaking is tiny. We calculate the scattering potentials, relaxation rates, and conductivity in this region and show that such emergent defect states are essential for the transport anisotropy observed in experiments.