Nematicity In FeSe Research Articles

Rigorous formalism for unconventional symmetry breaking in Fermi liquid theory and its application to nematicity in FeSe

Unconventional symmetry breaking due to nonlocal order parameters has attracted considerable attention in many strongly correlated metals. Famous examples are the nematic order in Fe-based superconductors and the star-of-David charge density order in kagome metals. Such exotic symmetry breaking in metals is a central issue of modern condensed matter physics, while its theoretical foundation is still unclear in comparison with the well-established theory of superconductivity. To overcome this difficulty, here we introduce the "form factor" that generalizes the nonlocal order parameter into the Luttinger-Ward (LW) Fermi liquid theory. We then construct a rigorous formalism of the "density-wave equation" that gives the thermodynamically stable form factor, similarly to the superconducting-gap equation. In addition, a rigorous expression of the Ginzburg-Landau free-energy for the unconventional order is presented to calculate various thermodynamic properties. In the next stage, we apply the derived formalism to a typical Fe-based superconductor FeSe, by using the one-loop LW function that represents the free-energy gain due to the interference among paramagnons. The following key experiments are naturally explained: (i) Lifshitz transition (=disappearance of an electron-pocket) due to the bond+orbital order below $T_c$. (ii) Curie-Weiss behavior of the nematic susceptibility at higher T, and the deviation from the Curie-Weiss behavior at lower T near the nematic quantum-critical-point. (iii) Scaling relation of the specific heat jump at $T_c$, $\Delta C/T_c \propto T_c^b$ with $b \sim 3$. (Note that b=0 in the BCS theory.) These results lead to a conclusion that the nematicity in FeSe is the bond+orbital order due to the "paramagnon interference mechanism". The present theory paves the way for solving various unconventional phase transition systems.

Frustrated magnetic interactions in FeSe

The structurally simplest high-temperature superconductor FeSe exhibits an intriguing superconducting nematic paramagnetic phase with unusual spin excitation spectra that are different from typical spin waves; thus, determining its effective magnetic exchange interactions is challenging. Here we report neutron scattering measurements of spin fluctuations of FeSe in the tetragonal paramagnetic phase. We show that the equal-time magnetic structure factor, $\mathcal{S}(\mathbf{Q})$, can be effectively modeled using the self-consistent Gaussian approximation calculation with highly frustrated nearest-neighbor (${J}_{1}$) and next-nearest-neighbor (${J}_{2}$) exchange couplings, and very weak further neighbor exchange interaction. Our results elucidate the frustrated magnetism in FeSe, which provides a natural explanation for the highly tunable superconductivity and nematicity in FeSe and related materials.

Stronger quantum fluctuation with larger spins: Emergent magnetism in the pressurized high-temperature superconductor FeSe

A counter-intuitive enhancement of quantum fluctuation with larger spins, together with a few novel physical phenomena, is discovered in studying the recently observed emergent magnetism in high-temperature superconductor FeSe under pressure. Starting with experimental crystalline structure from our high-pressure X-ray refinement, we analyze theoretically the stability of the magnetically ordered state with a realistic spin-fermion model. We find surprisingly that in comparison with the magnetically ordered Fe-pnictides, the larger spins in FeSe suffer even stronger long-range quantum fluctuation that diminishes their ordering at ambient pressure. This "fail-to-order" quantum spin liquid state then develops into an ordered state above 1GPa due to weakened fluctuation accompanying the reduction of anion height and carrier density. The ordering further benefits from the ferro-orbital order and shows the observed enhancement around 1GPa. We further clarify the controversial nature of magnetism and its interplay with nematicity in FeSe in the same unified picture for all Fe-based superconductors. In addition, the versatile itinerant carriers produce interesting correlated metal behavior in a large region of phase space. Our study establishes a generic exceptional paradigm of stronger quantum fluctuation with larger spins that complements the standard knowledge of insulating magnetism.

Diverse Exotic Orders and Fermiology in Fe-Based Superconductors: A Unified Mechanism for B1g/B2g Nematicity in FeSe/(Cs,Rb)Fe2As2 and Smectic Order in BaFe2As2

A rich variety of nematic/smectic orders in Fe-based superconductors is an important unsolved problem in strongly correlated electron systems. A unified understanding of these orders has been investigated for the last decade. In this study, we explain the B1g symmetry nematic transition in FeSe1−xTex, the B2g symmetry nematicity in AFe2As2 (A = Cs, Rb), and the smectic state in BaFe2As2 based on the same framework. We investigate the quantum interference mechanism between spin fluctuations by developing the density wave equation. The observed rich variety of nematic/smectic orders is naturally understood in this mechanism. The nematic/smectic orders depend on the characteristic shape and topology of the Fermi surface (FS) of each compound. 1) In FeSe1−xTex (nd = 6.0), each FS is very small and the dxy-orbital hole pocket is below the Fermi level. In this case, the small spin fluctuations on three dxz, dyz, and dxy orbitals cooperatively lead to the B1g nematic (q = 0) order without magnetization. The experimental Lifshitz transition below the nematic transition temperature (TS) is naturally reproduced. 2) In BaFe2As2 (nd = 6.0), the dxy-orbital hole pocket emerges around the M point, and each FS is relatively large. The strong spin fluctuations due to the dxy-orbital nesting give rise to the B1g nematic (q = 0) order and the smectic [q = (0, π)] order, and the latter transition temperature (T* ∼ 170K) exceeds the former one (TS ∼ 140K). 3) In heavily hole-doped AFe2As2 (nd = 5.5), the large dxy-orbital hole pocket and the four tiny Dirac pockets appear due to the hole-doping. The B2g nematic bond order emerges on the dxy-orbital hole pocket because of the same interference mechanism. The present paramagnon interference mechanism provides a unified explanation of why the variety of nematic/smectic orders in Fe-based superconductors is so rich, based on the well-established fermiology of Fe-based superconductors.

Spin-excitation anisotropy in the nematic state of detwinned FeSe

The origin of the electronic nematicity in FeSe is one of the most important unresolved puzzles in the study of iron-based superconductors. In both spin- and orbital-nematic models, the intrinsic magnetic excitations at Q1 = (1, 0) and Q2 = (0, 1) of twin-free FeSe are expected to provide decisive criteria for clarifying this issue. Although a spin-fluctuation anisotropy below 10 meV between Q1 and Q2 has been observed by inelastic neutron scattering at low temperature, it remains unclear whether such an anisotropy also persists at higher energies and associates with the nematic transition Ts. Here we use resonant inelastic X-ray scattering to probe the high-energy magnetic excitations of detwinned FeSe. A prominent anisotropy between the magnetic excitations along the H and K directions is found to persist to E ≈ 200 meV, which decreases gradually with increasing temperature and finally vanishes at a temperature around Ts. The measured high-energy spin excitations are dispersive and underdamped, which can be understood from a local-moment perspective.Taking together the large energy scale far beyond the dxz/dyz orbital splitting, we suggest that the nematicity in FeSe is probably spin-driven.

Optical Fingerprints of Nematicity in Iron-Based Superconductors

Nematicity, which refers to a phase of broken rotational but preserved translational symmetry, is underlined by the appearance of anisotropic properties and leaves remarkable fingerprints in all measurable physical quantities upon crossing the structural tetragonal-orthorhombic transition at Ts in several iron-based materials. Here, we review part of our own broadband optical investigations, addressing the impact of nematicity on the charge dynamics, as a function of temperature and of tunable applied stress, the latter acting as an external symmetry breaking field. We shall first focus our attention on FeSe, which undergoes a nematic (structural) transition without any subsequent onset of magnetic ordering below Ts. FeSe thus provides an opportunity to study nematicity without the limitations due to the reconstruction of the Fermi surface because of the spin-density-wave collective state in the orthorhombic phase, typical for several other iron-based superconductors. Our data reveal an astonishing anisotropy of the optical response in the mid-infrared-to-visible spectral range, which bears testimony of an important polarization of the underlying electronic structure in agreement with angle-resolved-photoemission-spectroscopy results. Our findings at high energy scales support models for the nematic phase resting on an orbital-ordering mechanism, supplemented by orbital selective band renormalization. The optical results at energies close to the Fermi level furthermore emphasize scenarios relying on scattering by anisotropic spin-fluctuations and shed new light on the origin of nematicity in FeSe. Moreover, the composition at which the associated Weiss temperature of the nematic susceptibility extrapolates to zero is found to be close to optimal doping (i.e., in coincidence with the largest superconducting transition temperature), boosting the debate to what extent nematic fluctuations contribute to the pairing-mechanism and generally affect the electronic structure of iron-based superconductors. The present review then offers a discussion of our optical data on the optimally hole-doped Ba0.6K0.4Fe2As2. We show that the stress-induced optical anisotropy in the infrared spectral range is reversible upon sweeping the applied stress and occurs only below the superconducting transition temperature. These findings demonstrate that there is a large electronic nematicity at optimal doping which extends right under the superconducting dome.

Evolution of transport properties in FeSe thin flakes with thickness approaching the two-dimensional limit

Electronic properties of FeSe can be tuned by various routes. Here, we present a comprehensive study on the evolution of the superconductivity and nematicity in FeSe with thickness from bulk single crystal down to bilayer ($\sim$ 1.1 nm) through exfoliation. With decreasing flake thickness, both the structural transition temperature $T_{\rm s}$ and the superconducting transition temperature $T_{\rm c}^{\rm zero}$ are greatly suppressed. The magnetic field ($B$) dependence of Hall resistance $R_{xy}$ at 15 K changes from $B$-nonlinear to $B$-linear behavior up to 9 T, as the thickness ($d$) is reduced to 13 nm. $T_{\rm c}$ is linearly dependent on the inverse of flake thickness (1/$d$) when $d\le$ 13 nm, and a clear drop of $T_{\rm c}$ appears with thickness smaller than 27 nm. The $I$-$V$ characteristic curves in ultrathin flakes reveal the signature of Berezinskii-Kosterlitz-Thouless (BKT) transition, indicating the presence of two-dimensional superconductivity. Anisotropic magnetoresistance measurements further support 2D superconductivity in few-layer FeSe. Increase of disorder scattering, anisotropic strains and dimensionality effect with reducing the thickness of FeSe flakes, might be taken into account for understanding these behaviors. Our study provides systematic insights into the evolution of the superconducting properties, structural transition and Hall resistance of a superconductor FeSe with flakes thickness and provides an effective way to find two-dimensional superconductivity as well as other 2D novel phenomena.

Relationship between Transport Anisotropy and Nematicity in FeSe

The mechanism behind the nematicity of FeSe is not known. Through elastoresitivity measurements it has been shown to be an electronic instability. However, so far measurements have extended only to small strains, where the response is linear. Here, we apply large elastic strains to FeSe, and perform two types of measurements. (1) Using applied strain to control twinning, the nematic resistive anisotropy at temperatures below the nematic transition temperature Ts is determined. (2) Resistive anisotropy is measured as nematicity is induced through applied strain at fixed temperature above Ts. In both cases, as nematicity strengthens the resistive anisotropy peaks about about 7%, then decreases. Below ~40 K, the nematic resistive anisotropy changes sign. We discuss possible implications of this behaviour for theories of nematicity. We report in addition: (1) Under experimentally accessible conditions with bulk crystals, stress, rather than strain, is the conjugate field to the nematicity of FeSe. (2) At low temperatures the twin boundary resistance is ~10% of the sample resistance, and must be properly subtracted to extract intrinsic resistivities. (3) Biaxial inplane compression increases both in-plane resistivity and the superconducting critical temperature Tc, consistent with a strong role of the yz orbital in the electronic correlations.

Singular magnetic anisotropy in the nematic phase of FeSe

FeSe is arguably the simplest, yet the most enigmatic, iron-based superconductor. Its nematic but non-magnetic ground state is unprecedented in this class of materials and stands out as a current puzzle. Here, our nuclear magnetic resonance measurements in the nematic state of mechanically detwinned FeSe reveal that both the Knight-shift and the spin–lattice relaxation rate 1/T1 possess an in-plane anisotropy opposite to that of the iron pnictides LaFeAsO and BaFe2As2. Using a microscopic electron model that includes spin–orbit coupling, our calculations show that an opposite quasiparticle weight ratio between the dxz and dyz orbitals leads to an opposite anisotropy of the orbital magnetic susceptibility, which explains our Knight-shift results. We attribute this property to a different nature of nematic order in the two compounds, predominantly bond type in FeSe and onsite ferro-orbital in pnictides. The T1 anisotropy is found to be inconsistent with existing neutron scattering data in FeSe, showing that the spin fluctuation spectrum reveals surprises at low energy, possibly from fluctuations that do not break C4 symmetry. Therefore, our results reveal that important information is hidden in these anisotropies and they place stringent constraints on the low-energy spin correlations as well as on the nature of nematicity in FeSe.

A first-principle perspective on electronic nematicity in FeSe

Electronic nematicity is an important order in most iron-based superconductors, and FeSe represents a special example, in which nematicity disentangles from spin ordering. A first-principle description of this order remains elusive. Here, we show that by carefully searching the paramagnetic energy landscape within the density functional theory, a nematic solution stands out at either the +U or hybrid functional level with the lowest energy. The band structure and Fermi surface can be well compared with the recent experimental results. Symmetry analysis assigns the dominant order parameter to the Eu irreducible representations of the D4h point group. Distinct from the B1g Ising nematicity as widely discussed in the context of vestigial stripe antiferromagnetic order, the two-component Eu vector order features mixing of the Fe d-orbitals and inversion symmetry breaking, which lead to striking experimental consequences, e.g., missing of an electron pocket.

Magnetic tricritical point and nematicity in FeSe under pressure

Magnetism induced by external pressure ($p$) was studied in a FeSe crystal sample by means of muon-spin rotation. The magnetic transition changes from second-order to first-order for pressures exceeding the critical value $p_{{\rm c}}\simeq2.4-2.5$ GPa. The magnetic ordering temperature ($T_{{\rm N}}$) and the value of the magnetic moment per Fe site ($m_{{\rm Fe}}$) increase continuously with increasing pressure, reaching $T_{{\rm N}}\simeq50$~K and $m_{{\rm Fe}}\simeq0.25$ $\mu_{{\rm B}}$ at $p\simeq2.6$ GPa, respectively. No pronounced features at both $T_{{\rm N}}(p)$ and $m_{{\rm Fe}}(p)$ are detected at $p\simeq p_{{\rm c}}$, thus suggesting that the stripe-type magnetic order in FeSe remains unchanged above and below the critical pressure $p_{{\rm c}}$. A phenomenological model for the $(p,T)$ phase diagram of FeSe reveals that these observations are consistent with a scenario where the nematic transitions of FeSe at low and high pressures are driven by different mechanisms.

NMR evidence for static local nematicity and its cooperative interplay with low-energy magnetic fluctuations in FeSe under pressure

We present $^{77}$Se-NMR measurements on single-crystalline FeSe under pressures up to 2 GPa. Based on the observation of the splitting and broadening of the NMR spectrum due to structural twin domains, we discovered that static, local nematic ordering exists well above the bulk nematic ordering temperature, $T_{\rm s}$. The static, local nematic order and the low-energy stripe-type antiferromagnetic spin fluctuations, as revealed by NMR spin-lattice relaxation rate measurements, are both insensitive to pressure application. These NMR results provide clear evidence for the microscopic cooperation between magnetism and local nematicity in FeSe.

Robust short-range-ordered nematicity in FeSe evidenced by high-pressure NMR

We report high-pressure $^{77}$Se NMR studies on FeSe single crystals that reveal a prominent inhomogeneous NMR linewidth broadening upon cooling, with the magnetic field applied along the tetragonal [110] direction. The data indicate the existence of short-range-ordered, inhomogeneous electronic nematicity, which has surprisingly long time scales over milliseconds. The short-range order survives temperatures up to $8$ times the structural transition temperature, and remains robust against pressure, in contrast to the strong pressure-dependence of the orbital ordering, structural transition, and the ground state magnetism. Such an extended region of static nematicity in the ($P$,$T$) space of FeSe indicates an enormously large fluctuating regime, and provide fresh insights and constraints to the understanding of electronic nematicity in iron-based superconductors.

Unveiling the hidden nematicity and spin subsystem in FeSe

The nematic order (nematicity) is considered as one of the essential ingredients to understand the mechanism of Fe-based superconductivity. In most Fe-based superconductors (pnictides), nematic order is reasonably close to the antiferromagnetic order. In FeSe, in contrast, a nematic order emerges below the structure phase transition at Ts = 90 K with no magnetic order. The case of FeSe is of paramount importance to a universal picture of Fe-based superconductors. The polarized ultrafast spectroscopy provides a tool to probe simultaneously the electronic structure and the magnetic interactions through quasiparticle dynamics. Here we show that this approach reveals both the electronic and magnetic nematicity below and, surprisingly, its fluctuations far above Ts to at least 200 K. The quantitative pump–probe data clearly identify a correlation between the topology of the Fermi surface and the magnetism in all temperature regimes, thus providing profound insight into the driving factors of nematicity in FeSe and the origin of its uniqueness.

Effects of strain on the electronic structure, superconductivity, and nematicity in FeSe studied by angle-resolved photoemission spectroscopy

One of central issues in iron-based superconductors is the role of structural change to the superconducting transition temperature (T_c). It was found in FeSe that the lattice strain leads to a drastic increase in T_c, accompanied by suppression of nematic order. By angle-resolved photoemission spectroscopy on tensile- or compressive-strained and strain-free FeSe, we experimentally show that the in-plane strain causes a marked change in the energy overlap (DeltaE_{h-e}) between the hole and electron pockets in the normal state. The change in DeltaE_{h-e} modifies the Fermi-surface volume, leading to a change in T_c. Furthermore, the strength of nematicity is also found to be characterized by DeltaE_{h-e}. These results suggest that the key to understanding the phase diagram is the fermiology and interactions linked to the semimetallic band overlap.

Pressure Induced Stripe-Order Antiferromagnetism and First-Order Phase Transition in FeSe.

To elucidate the magnetic structure and the origin of the nematicity in FeSe, we perform a high-pressure ^{77}Se NMR study on FeSe single crystals. We find a suppression of the structural transition temperature with pressure up to about 2GPa from the anisotropy of the Knight shift. Above 2GPa, a stripe-order antiferromagnetism that breaks the spatial fourfold rotational symmetry is determined by the NMR spectra under different field orientations and with temperatures down to 50mK. The magnetic phase transition is revealed to be first-order type, implying the existence of a concomitant structural transition via a spin-lattice coupling. Stripe-type spin fluctuations are observed at high temperatures, and remain strong with pressure. These results provide clear evidence for strong coupling between nematicity and magnetism in FeSe, and therefore support a universal scenario of magnetic driven nematicity in iron-based superconductors.

Effect of nematic ordering on electronic structure of FeSe.

Electronically driven nematic order is often considered as an essential ingredient of high-temperature superconductivity. Its elusive nature in iron-based superconductors resulted in a controversy not only as regards its origin but also as to the degree of its influence on the electronic structure even in the simplest representative material FeSe. Here we utilized angle-resolved photoemission spectroscopy and density functional theory calculations to study the influence of the nematic order on the electronic structure of FeSe and determine its exact energy and momentum scales. Our results strongly suggest that the nematicity in FeSe is electronically driven, we resolve the recent controversy and provide the necessary quantitative experimental basis for a successful theory of superconductivity in iron-based materials which takes into account both, spin-orbit interaction and electronic nematicity.

Orbital-dependent Fermi surface shrinking as a fingerprint of nematicity in FeSe

The large anisotropy in the electronic properties across a structural transition in several correlated systems has been identified as the key manifestation of electronic nematic order, breaking rotational symmetry. In this context, FeSe is attracting tremendous interest, since electronic nematicity develops over a wide range of temperatures, allowing accurate experimental investigation. Here we combine angle-resolved photoemission spectroscopy and theoretical calculations based on a realistic multi-orbital model to unveil the microscopic mechanism responsible for the evolution of the electronic structure of FeSe across the nematic transition. We show that the self-energy corrections due to the exchange of spin fluctuations between hole and electron pockets are responsible for an orbital-dependent shrinking of the Fermi Surface that affects mainly the $xz/yz$ parts of the Fermi surface. This result is consistent with our experimental observation of the Fermi Surface in the high-temperature tetragonal phase, that includes the $xy$ electron sheet that was not clearly resolved before. In the low-temperature nematic phase, we experimentally confirm the appearance of a large ($\sim$ 50meV) $xz/yz$ splittings. It can be well reproduced in our model by assuming a moderate splitting between spin fluctuations along the $x$ and $y$ crystallographic directions. Our mechanism shows how the full entanglement between orbital and spin degrees of freedom can make a spin-driven nematic transition equivalent to an effective orbital order.

Strong cooperative coupling of pressure-induced magnetic order and nematicity in FeSe.

A hallmark of the iron-based superconductors is the strong coupling between magnetic, structural and electronic degrees of freedom. However, a universal picture of the normal state properties of these compounds has been confounded by recent investigations of FeSe where the nematic (structural) and magnetic transitions appear to be decoupled. Here, using synchrotron-based high-energy x-ray diffraction and time-domain Mössbauer spectroscopy, we show that nematicity and magnetism in FeSe under applied pressure are indeed strongly coupled. Distinct structural and magnetic transitions are observed for pressures between 1.0 and 1.7 GPa and merge into a single first-order transition for pressures ≳1.7 GPa, reminiscent of what has been found for the evolution of these transitions in the prototypical system Ba(Fe1−xCox)2As2. Our results are consistent with a spin-driven mechanism for nematic order in FeSe and provide an important step towards a universal description of the normal state properties of the iron-based superconductors.

Charge-induced nematicity in FeSe

The spontaneous appearance of nematicity, a state of matter that breaks rotation but not translation symmetry, is one of the most intriguing properties of the iron-based superconductors (Fe SC), and has relevance for the cuprates as well. Establishing the critical electronic modes behind nematicity remains a challenge, however, because their associated susceptibilities are not easily accessible by conventional probes. Here, using FeSe as a model system, and symmetry-resolved electronic Raman scattering as a probe, we unravel the presence of critical charge nematic fluctuations near the structural/nematic transition temperature, [Formula: see text] 90 K. The diverging behavior of the associated nematic susceptibility foretells the presence of a Pomeranchuk instability of the Fermi surface with d-wave symmetry. The excellent scaling between the observed nematic susceptibility and elastic modulus data demonstrates that the structural distortion is driven by this d-wave Pomeranchuk transition. Our results make a strong case for charge-induced nematicity in FeSe.