Checkerboard Charge Order Research Articles

Checkerboard ordered state in a superconducting FeSe/SrTiO3(001) monolayer

Ordered electronic states have been extensively explored in cuprates and iron-based unconventional superconductors, but seldom observed in the epitaxial $\mathrm{FeSe}/\mathrm{SrTi}{\mathrm{O}}_{3}(001)$ monolayer (FeSe/STO) with an enhanced superconducting transition temperature $({T}_{c})$. Here, by using scanning tunneling microscopy/scanning tunneling spectroscopy (STM/STS), we reveal a checkerboard charge order in the epitaxial FeSe/STO monolayer, with a period of four times the inter-iron-atom distance along two perpendicular directions of the iron lattice. This ordered state is uniquely present in the superconducting FeSe/STO monolayer, even at liquid nitrogen temperature, but is absent in the nonsuperconducting FeSe monolayer or bilayer. Quasi-particle interference measurements further confirm it as a static order without an energy-dependent dispersion and is gapped out within the superconductivity gap. The intensity of the charge order shows an enhancement near the superconducting transition temperature, thus implying a correlation with the $\text{high-}{T}_{c}$ superconductivity in the FeSe/STO monolayer. This study provides a new basis for exploring the ordered electronic states and their interplay with $\text{high-}{T}_{c}$ superconductivity in the FeSe monolayer.

Charge order driven by multiple-Q spin fluctuations in heavily electron-doped iron selenide superconductors

Intertwined spin and charge orders have been widely studied in high-temperature superconductors, since their fluctuations may facilitate electron pairing; however, they are rarely identified in heavily electron-doped iron selenides. Here, using scanning tunneling microscopy, we show that when the superconductivity of (Li0.84Fe0.16OH)Fe1-xSe is suppressed by introducing Fe-site defects, a short-ranged checkerboard charge order emerges, propagating along the Fe-Fe directions with an approximately 2aFe period. It persists throughout the whole phase space tuned by Fe-site defect density, from a defect-pinned local pattern in optimally doped samples to an extended order in samples with lower Tc or non-superconducting. Intriguingly, our simulations indicate that the charge order is likely driven by multiple-Q spin density waves originating from the spin fluctuations observed by inelastic neutron scattering. Our study proves the presence of a competing order in heavily electron-doped iron selenides, and demonstrates the potential of charge order as a tool to detect spin fluctuations.

Hidden Charge Order in an Iron Oxide Square-Lattice Compound.

Since the discovery of charge disproportionation in the FeO_{2} square-lattice compound Sr_{3}Fe_{2}O_{7} by Mössbauer spectroscopy more than fifty years ago, the spatial ordering pattern of the disproportionated charges has remained "hidden" to conventional diffraction probes, despite numerous x-ray and neutron scattering studies. We have used neutron Larmor diffraction and Fe K-edge resonant x-ray scattering to demonstrate checkerboard charge order in the FeO_{2} planes that vanishes at a sharp second-order phase transition upon heating above 332K. Stacking disorder of the checkerboard pattern due to frustrated interlayer interactions broadens the corresponding superstructure reflections and greatly reduces their amplitude, thus explaining the difficulty of detecting them by conventional probes. We discuss the implications of these findings for research on "hidden order" in other materials.

Wang-MacDonald d -Wave Vortex Cores Observed in Heavily Overdoped Bi2Sr2CaCu2O8+δ

Low-magnetic-field scanning tunneling spectroscopy of individual Abrikosov vortices in heavily overdoped Bi2Sr2CaCu2O8+δ unveils a clear d-wave electronic structure of the vortex core, with a zero-bias conductance peak at the vortex center that splits with increasing distance from the core. We show that previously reported unconventional electronic structures, including the low-energy checkerboard charge order in the vortex halo and the absence of a zero-bias conductance peak at the vortex center, are direct consequences of short intervortex distance and consequent vortex-vortex interactions prevailing in earlier experiments.Received 16 February 2021Revised 29 April 2021Accepted 17 June 2021DOI:https://doi.org/10.1103/PhysRevX.11.031040Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasLocal density of statesVortices in superconductorsd-wavePhysical SystemsHigh-temperature superconductorsTechniquesScanning tunneling microscopyScanning tunneling spectroscopyCondensed Matter, Materials & Applied Physics

Magnetic‐Field Tunable Intertwined Checkerboard Charge Order and Nematicity in the Surface Layer of Sr2RuO4 (Adv. Mater. 32/2021)

Magnetic‐Field Tunable Intertwined Checkerboard Charge Order and Nematicity in the Surface Layer of Sr₂RuO₄ (Adv. Mater. 32/2021)

Magnetic-Field Tunable Intertwined Checkerboard Charge Order and Nematicity in the Surface Layer of Sr2 RuO4.

In strongly correlated electron materials, the electronic, spin, and charge degrees of freedom are closely intertwined. This often leads to the stabilization of emergent orders that are highly sensitive to external physical stimuli promising opportunities for technological applications. In perovskite ruthenates, this sensitivity manifests in dramatic changes of the physical properties with subtle structural details of the RuO6 octahedra, stabilizing enigmatic correlated ground states, from a hotly debated superconducting state via electronic nematicity and metamagnetic quantum criticality to ferromagnetism. Here, it is demonstrated that the rotation of the RuO6 octahedra in the surface layer of Sr2 RuO4 generates new emergent orders not observed in the bulk material. Through atomic-scale spectroscopic characterization of the low-energy electronic states, four van Hove singularities are identified in the vicinity of the Fermi energy. The singularities can be directly linked to intertwined nematic and checkerboard charge order. Tuning of one of these van Hove singularities by magnetic field is demonstrated, suggesting that the surface layer undergoes a Lifshitz transition at a magnetic field of ≈32T. The results establish the surface layer of Sr2 RuO4 as an exciting 2D correlated electron system and highlight the opportunities for engineering the low-energy electronic states in thesesystems.

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.

Suppression of the outwards-dispersing branches in hour-glass magnetic spectra induced by nanoscale phase separation in La2−xSrxCoO4

Here, we reinvestigate the underlying charge and magnetic correlations in ${\mathrm{La}}_{2\ensuremath{-}x}{\mathrm{Sr}}_{x}{\mathrm{CoO}}_{4}$ within a combined study of muon spin relaxation ($\ensuremath{\mu}\mathrm{SR}$), x-ray photon correlation spectroscopy, synchrotron radiation single-crystal x-ray diffraction, and neutron-scattering measurements. For ${\mathrm{La}}_{2\ensuremath{-}x}{\mathrm{Sr}}_{x}{\mathrm{CoO}}_{4}$ around 1/3 hole-doping, the significant charge correlations that are responsible for the emergence of hour-glass magnetic spectra are found to be slowly fluctuating checkerboard charge ordering (CBCO) correlations with an onset temperature being independent of hole-doping. This reveals a local origin of the CBCO correlations in the rather ionic cobaltates, thus being distinct from the more covalent nickelates. Moreover, we report the observation of a very similar temperature dependence of the intensities of in-plane high- and low-energy excitations within the ${\mathrm{La}}_{2\ensuremath{-}x}{\mathrm{Sr}}_{x}{\mathrm{CoO}}_{4}$ hour-glass spectrum. This observation shows that nano phase separation is distinct from conventional phase separation and that there is a coupling between nanometer-sized undoped and hole-doped islands. Based on this, it is possible to propose a microscopic scenario for the suppression of the outwards-dispersing branches in hour-glass magnetic spectra.

Langevin approach to lattice dynamics in a charge-ordered polaronic system

We use a Langevin approach to treat the finite temperature dynamics of displacement variables in the half-filled spinless Holstein model. Working in the adiabatic regime we exploit the smallness of the adiabatic parameter to simplify the memory effects and estimate displacement costs from an "instantaneous" electronic Hamiltonian. We use a phenomenological damping rate, and uncorrelated thermal noise. The low temperature state has checkerboard charge order (CO) and the Langevin scheme generates equilibrium thermodynamic properties that accurately match Monte Carlo results. It additionally yields the dynamical structure factor, $D({\bf q}, \omega)$, from the displacement field $x({\bf r}, t)$. We observe four regimes with increasing temperature, $T$, classified in relation to the charge ordering temperature, $T_c$, and the `polaron formation' temperature $T_P$, with $ T_c \ll T_P$. For $T \ll T_c$ the oscillations are harmonic, leading to dispersive phonons, with increasing $T$ bringing in anharmonic, momentum dependent, corrections. For $T \sim T_c$, thermal tunneling events of the $x({\bf r})$ field occur, with a propagating `domain' pattern at wavevector ${\bf q} \sim (\pi, \pi)$ and low energy weight in $D({\bf q}, \omega)$. When $T_c < T < T_P$, the disordered polaron regime, domain structures vanish, the dispersion narrows, and low energy weight is lost. For $T \gtrsim T_P$ we essentially have uncorrelated local oscillations. We propose simple models to analyse this rich dynamics.

Emergent charge order near the doping-induced Mott-insulating quantum phase transition in Sr3Ru2O7

Search for novel electronically ordered states of matter emerging near quantum phase transitions is an intriguing frontier of condensed matter physics. In ruthenates, the interplay between Coulomb correlations among the 4d electronic states and their spin-orbit interactions, lead to complex forms of electronic phenomena. Here we investigate the double layered Sr3(Ru1−xMnx)2O7 and its doping-induced quantum phase transition from a metal to an antiferromagnetic Mott insulator. Using spectroscopic imaging with the scanning tunneling microscope, we visualize the evolution of the electronic states in real- and momentum-space. We find a partial-gap at the Fermi energy that develops with doping to form a weak Mott insulating state. Near the quantum phase transition, we discover a spatial electronic reorganization into a commensurate checkerboard charge order. These findings bear a resemblance to the universal charge order in the pseudogap phase of cuprates and demonstrate the ubiquity of charge order that emanates from doped Mott insulators.

Visualization of the periodic modulation of Cooper pairing in a cuprate superconductor

A major obstacle in understanding the mechanism of Cooper pairing in the cuprates is the existence of various intertwined orders associated with spin, charge, and Cooper pairs. Of particular importance is the ubiquitous charge order features that have been observed in a variety of cuprates, especially in the underdoped regime of the phase diagram. To explain the origin of the charge order and its implication to the superconducting phase, many theoretical models have been proposed, such as charge stripes, electronic nematicity, and Fermi surface instability. A highly appealing physical picture is the so-called pair density wave (PDW), a periodic modulation of Cooper paring in space, which may also induce a charge order. To elucidate the existence and nature of the PDW order, here we use scanning tunneling microscopy (STM) to investigate a severely underdoped Bi2Sr2CaCu2O8+{\delta}, in which superconductivity just emerges on top of a pronounced checkerboard charge order. By analyzing the spatial distribution of the spectral features characteristic of superconductivity, we observe a periodic modulation of both the superconducting coherence peak and gap depth, demonstrating the existence of a density wave order of Cooper pairing. The PDW order has the same spatial periodicity as the charge order, and the amplitudes of the two orders exhibit clear positive correlation. These results shed important new lights on the origin of and interplay between the charge order and Cooper pairing modulation in the cuprates.

1/8 Anomaly Induced by the Substitution of Dy in Underdoped Bi2Sr2Ca1−x Dy x Cu2O8+δ

An anomalous “dip” feature near the hole concentration of p ∼ 1/8 on the T c − p curve of Bi2Sr2 Ca1−x Dy x Cu2O8+δ (Dy-Bi2212) was reported. Dy-Bi2212 with p∼ 1/8 exhibited a “checkerboard charge order” with a wave vector ∼ 0.3 r.l.u., and well-defined coherent peaks were not observed in the quasiparticle tunneling spectra of this material, obtained from scanning tunneling microscopy/spectroscopy (STM/STS) experiments. The dip of the T c−p curve induced by the substitution of Dy in Bi2212 was similar to that observed in Zn-doped Bi2212 (Akoshima et al., Phys. Rev. B 57, 7491 1998), suggesting that the suppression of T c near p∼ 1/8 in Dy-Bi2212 is related to the pinning of the charge order.

Photoinduced ultrafast charge-order melting: Charge-order inversion and nonthermal effects

The effect of photoexcitation is studied for a system with checkerboard charge order induced by displacements of ligands around a metal site. The motion of the ligands is treated classically and the electronic charges are simplified to two-level molecular bond charges. The calculations are done for a checkerboard charge-ordered system with about 100 000 ligand oscillators coupled to a fixed-temperature bath. The initial photoexcitation is followed by a rapid decrease in the charge-order parameter within 50–100 femtoseconds while leaving the correlation length almost unchanged. Depending on the fluence, a complete melting of the charge order occurs in less than a picosecond. While for low fluences, the system returns to its original state, for full melting, it recovers to its broken-symmetry state leading to an inversion of the charge order. Finally, for small long-range interactions, recovery can be slow due to domain formation.

Charge ordering inNi1+/Ni2+nickelates:La4Ni3O8andLa3Ni2O6

Ab initio calculations have been performed to unravel the origin of the recently found superlattice peaks in the trilayer nickelate La$_4$Ni$_3$O$_8$. These peaks arise from static charge ordering of Ni$^{2+}$/ Ni$^{1+}$ stripes oriented at 45$^{\circ}$ to the Ni-O bonds. An insulating state originates from a combination of structural distortions and magnetic order, with the gap being formed solely within the d$_{x^2-y^2}$ manifold of states. When doped, electrons or holes would go into these states, in a similar fashion to what occurs in the cuprates. Analogous calculations suggest that checkerboard charge order should occur in the bilayer nickelate La$_3$Ni$_2$O$_6$. These results reveal a close connection between La$_4$Ni$_3$O$_8$ and La$_3$Ni$_2$O$_6$ with La$_{2-x}$Sr$_x$NiO$_4$ for x=1/3 and x=1/2, respectively.

Visualizing the evolution from the Mott insulator to a charge-ordered insulator in lightly doped cuprates

A central question in the high temperature cuprate superconductors is the fate of the parent Mott insulator upon charge doping. Here we use scanning tunneling microscopy to investigate the local electronic structure of lightly doped cuprate in the antiferromagnetic insulating regime. We show that the doped charge induces a spectral weight transfer from the high energy Hubbard bands to the low energy in-gap states. With increasing doping, a V-shaped density of state suppression occurs at the Fermi level, which is accompanied by the emergence of checkerboard charge order. The new STM perspective revealed here is the cuprates first become a charge ordered insulator upon doping. Subsequently, with further doping, Fermi surface and high temperature superconductivity grow out of it.

Out-of-Plane Disorder Effects on the Energy Gaps and Electronic Charge Order in Bi2Sr1.7R0.3CuO6+δ (R = La and Eu)

We performed STM/STS experiments at 8 K in Bi2Sr1.7R0.3CuO6+δ (R-Bi2201) systems that had optimal (OP) hole-doping levels (∼0.17) but different Tc values, 35 K for R = La and 20 K for R = Eu, and examined out-of-plane disorder effects on the superconducting (SC) gap (SCG) and the pseudogap (PG) which is associated with the so-called “checkerboard charge order” (CCO). As out-of-plane disorders are strengthened by replacing La with Eu in OP R-Bi2201, the antinodal PG size ΔPG increases from ∼30 to ∼60 meV, the nodal SCG size ΔSC seems to decrease from ∼7 to ∼4 meV, and the pairing gap amplitude Δ0 or d-wave gap size at the antinodes is almost unchanged (Δ0 ∼ 15 meV). These gap sizes for OP doping in Eu-Bi2201 are comparable to those for an underdoping level of ∼0.1 in La-Bi2201. Although out-of-plane disorders strongly affect the electronic system of the Cu–O plane, they have no effect on the period of the CCO, which is five times the lattice constant (5a) along the Cu–O bond directions for OP doping and 4a...

Metallic ferromagnetism-insulating charge order transition in doped manganites

We show that an interplay of double exchange and impurity randomness can explain the competition between metal-ferromagnetic and insulating charge ordered states in doped manganites. The double exchange is simplified in the Ising type, whereas the randomness is modeled by the Falicov-Kimball binary distribution. The combined model is considered in a framework of dynamical mean-field theory. Using the Kubo-Greenwood formalism, the transport coefficients are explicitly expressed in terms of single particle spectral functions. Dividing the system into two sublattices we have pointed out a direct calculation to the checkerboard charge order parameter and the magnetizations. Numerical results show us that the checkerboard charge order can settle inside the ferromagnetic state at low temperature. An insulator-metal transition is also found at the point of the checkerboard charge order-ferromagnetic transition.

Inelastic neutron scattering study of phonon anomalies in La1.5Sr0.5NiO4

The high-energy phonons in La1.5Sr0.5NiO4, in which the checkerboard charge ordering is formed, was investigated by the inelastic neutron scattering. We found that the longitudinal modes show strong anomalies compared with La2NiO4. We argue the similarity and difference in the phonon anomalies between the present sample and the preceding works of different compositions.

Impact of local-moment fluctuations on the magnetic degeneracy of iron arsenide superconductors

We investigate the fate of the orthorhombic stripe-type magnetic state (ordering vectors $\left(\pi,0\right)$/$\left(0,\pi\right)$), observed in most iron-pnictide superconductors, in the presence of localized magnetic moments that tend to form a Neel state (ordering vector $\left(\pi,\pi\right)$). We show that before long-range Neel order sets in, the coupling between the conduction electrons and the fluctuations of the local moments favors an unusual magnetic state consisting of a coherent superposition of the $\left(\pi,0\right)$ and $\left(0,\pi\right)$ orders that preserves tetragonal symmetry. The magnetization of this state is non-uniform and induces a simultaneous checkerboard charge order. We discuss signatures of this magnetic configuration on the electronic spectrum and its impact on the superconducting state, showing that its phase space for coexistence with the $s^{+-}$ state is smaller than the stripe-type state. Our results shed light on recent experimental observations on $\mathrm{Ba(Fe_{1-x}Mn_{x})_{2}As_{2}}$ compounds, where the Neel-type local Mn moments interact with the Fe conduction electrons.

Charge order and possible bias-induced metastable state in the organic conductor β-(meso-DMBEDT-TTF)2PF6: effects of structural distortion

We theoretically investigate charge order and nonlinear conduction in the quasi-two-dimensional organic conductor β-(meso-DMBEDT-TTF)2PF6 (DMBEDT-TTF=dimethylbis(ethylenedithio)tetrathiafulvalene). Within the Hartree–Fock approximation, we study the effects of structural distortion on the experimentally observed checkerboard charge order and its bias-induced melting by using an extended Hubbard model with Peierls- and Holstein-types of electron–lattice interactions. The structural distortion is important in realizing the charge order. The current–voltage characteristics obtained by a nonequilibrium Green’s function method indicate that a charge-ordered insulating state changes into a conductive state. Although the charge order and lattice distortions are largely suppressed at a threshold voltage, they remain finite even in the conductive state. We discuss the relevance of the results to experimental observations, especially to a possible bias-induced metastable state.