Orbital Fluctuations Research Articles

Raman scattering in FeSe-based superconductors

Raman scattering is one of the fundamental techniques in modern condensed matter physics. Its technical uniqueness allows to probe a variety of excitations in condensed matter and their couplings. FeSe-based superconductors have abundant structural phases like bulk, monolayer, Fe-deficient phases, and intercalated phases. Meanwhile, they exhibit rich electronic and magnetic phase diagrams and easily tunable superconducting transition temperatures ranged from a few Kelvins up to more than 40 K. This leads to the formation of an ideal research platform for superconductivity and has attracted a great deal of interest in the field. In this paper, we made a brief introduction to the structural phases, magnetism and electronic structures of FeSe-based superconductors. And we focused on the recent progress on Raman scattering studies of FeSe-based superconductors. For phononic Raman scattering, we introduced the Raman scattering measurement of the phonons in the Fe(Se,Te) system firstly. Next, in order to further explore the influence of doping on T c, we discussed the doping/substitution of A-sites and Fe-sites of A x Fe2− y Se2. Finally, we studied the correlation between superconductivity and various structures. We then discussed two-magnon and spin-phonon coupling in the materials. We discussed the relationship between magnetic order and superconductivity, and the effect of magnetic order on phonons. For electronic Raman response, we presented the discussions and analysis on electronic nematic phases. We introduced the observation of nematic phases by different measurement methods. And we discussed the origin of nematic phases: spin fluctuations or/and orbital (charge) fluctuations, different measurement methods have different conclusions. Finally, we introduced the studies in monolayer FeSe and some Raman scattering results there. We introduced the discovery of T c above 77 K in monolayer FeSe/SrTiO3. And we discussed many novel physical properties such as the opening of energy gap in 1UC FeSe/NbSTO and the charge transfer and ferroelectric phase transition between 1-UC thin film and NbSTO substrate.

Fermi Surface, Pressure-Induced Antiferromagnetic Order, and Superconductivity in FeSe

The pressure dependence of the structural ($T_s$), antiferromagnetic ($T_m$), and superconducting ($T_c$) transition temperatures in FeSe is investigated on the basis of the 16-band $d$-$p$ model. At ambient pressure, a shallow hole pocket disappears due to the correlation effect, as observed in the angular-resolved photoemission spectroscopy (ARPES) and quantum oscillation (QO) experiments, resulting in the suppression of the antiferromagnetic order, in contrast to the other iron pnictides. The orbital-polarization interaction between the Fe $d$ orbital and Se $p$ orbital is found to drive the ferro-orbital order responsible for the structural transition without accompanying the antiferromagnetic order. The pressure dependence of the Fermi surfaces is derived from the first-principles calculation and is found to well account for the opposite pressure dependences of $T_s$ and $T_m$, around which the enhanced orbital and magnetic fluctuations cause the double-dome structure of the eigenvalue $\lambda$ in the Eliashberg equation, as consistent with that of $T_c$ in FeSe.

Orbitally-driven Mid-Burdigalian Coastal Sabkha cycles from the Sivas Basin: Sedimentological, paleontological, and geochemical data

The Early to Middle Miocene period in the Sivas Basin is represented by a thick deposition of mud-dominated marine sediments (Karacaören Formation) in an overall bay-like paleogeographic setting. Previous studies have suggested that halokinesis-induced local tectonics and orbital climatic fluctuations have been the primary controls on depositional evolution. In the present study, we investigated a 60-m–thick Karaman Gypsum Member (KGM), located at the base of the Karacaören Formation, in terms of its sedimentological, paleontological; both palynology and nannofossil biostratigraphy, and geochemical/mineralogical aspects to better characterize climatic controls on its deposition. Nannofossil assemblages indicate that the studied member is confined within the Sphenolithus belemnos Zone (NN3; 18.92–17.97Ma), and is mainly composed of rhythmic alternations of supratidal evaporites and lagoonal/marine mudstones. The palynological data demonstrate a minor, yet consistent, occurrence of megathermic taxa and numerous dinoflagellate cysts in mudstone intervals. The high chemical proxy of alteration (CPA) values and occurrence of kaolinites in the same mudstone intervals further prove deposition under warm and humid subtropical conditions. The perfect match between the environmental cyclicities in the KGM with a benthic δ18O curve justifies a strong control of obliquity/short eccentricity of a 40–100kyr duration, and hints at radical precipitation changes throughout the individual orbital cycles. These precipitation changes are explained by extreme latitudinal precipitation gradients during relatively hot periods before the mid-Miocene Climatic Optimum.

Transverse acoustic phonon anomalies at intermediate wave vectors in MgV2O4

Magnetic spinels (with chemical formula $AX_{2}$O$_{4}$, with $X$ a 3$d$ transition metal ion) that also have an orbital degeneracy are Jahn-Teller active and hence possess a coupling between spin and lattice degrees of freedom. At high temperatures, MgV$_{2}$O$_{4}$ is a cubic spinel based on V$^{3+}$ ions with a spin $S$=1 and a triply degenerate orbital ground state. A structural transition occurs at T$_{OO}$=63 K to an orbitally ordered phase with a tetragonal unit cell followed by an antiferromagnetic transition of T$_{N}$=42 K on cooling. We apply neutron spectroscopy in single crystals of MgV$_{2}$O$_{4}$ to show an anomaly for intermediate wavevectors at T$_{OO}$ associated with the acoustic phonon sensitive to the shear elastic modulus $\left(C_{11}-C_{12}\right)/2$. On warming, the shear mode softens for momentum transfers near close to half the Brillouin zone boundary, but recovers near the zone centre. High resolution spin-echo measurements further illustrate a temporal broadening with increased temperature over this intermediate range of wavevectors, indicative of a reduction in phonon lifetime. A subtle shift in phonon frequencies over the same range of momentum transfers is observed with magnetic fields. We discuss this acoustic anomaly in context of coupling to orbital and charge fluctuations.

Nematicity, magnetism, and superconductivity in FeSe under pressure: Unified explanation based on the self-consistent vertex correction theory

To understand the rich electronic phase diagram in FeSe under pressure that vividly demonstrates the strong interplay between the nematicity, magnetism and superconductivity, we analyze the electronic states by including the higher-order many-body effects called the vertex correction (VC). We predict the pressure-induced emergence of xy-orbital hole-pocket based on the first-principles analysis. Due to this pressure-induced Lifshitz transition, the spin fluctuations on the xy orbital are enhanced, whereas those on xz,yz orbitals are gradually reduced. For this reason, nonmagnetic orbital order $O = n_{xz} - n_{yz}$, which is driven by the spin fluctuations on xz,yz orbitals through the intra-orbital VCs, is suppressed, and it is replaced with the magnetism of xy-orbital d-electrons. The nodal s-wave state at ambient pressure ($O > 0$) and the enhancement of Tc under pressure are driven by the cooperation between the spin and orbital fluctuations.

Excitations and relaxation dynamics in multiferroic GeV4S8 studied by terahertz and dielectric spectroscopy

We report on THz time-domain spectroscopy on multiferroic ${\mathrm{GeV}}_{4}{\mathrm{S}}_{8}$, which undergoes orbital ordering at a Jahn-Teller transition at 30.5 K and exhibits antiferromagnetic order below 14.6 K. The THz experiments are complemented by dielectric experiments at audio and radio frequencies. We identify a low-lying excitation close to 0.5 THz, which is only weakly temperature dependent and probably corresponds to a molecular excitation within the electronic level scheme of the ${\mathrm{V}}_{4}$ clusters. In addition, we detect complex temperature-dependent behavior of a low-lying phononic excitation, closely linked to the onset of orbitally driven ferroelectricity. In the high-temperature cubic phase, which is paramagnetic and orbitally disordered, this excitation is of relaxational character becomes an overdamped Lorentzian mode in the orbitally ordered phase below the Jahn-Teller transition, and finally appears as well-defined phonon excitation in the antiferromagnetic state. Abrupt changes in the real and imaginary parts of the complex dielectric permittivity show that orbital ordering appears via a structural phase transition with strong first-order character and that the onset of antiferromagnetic order is accompanied by significant structural changes, which are of first-order character, too. Dielectric spectroscopy documents that at low frequencies, significant dipolar relaxations are present in the orbitally ordered, paramagnetic phase only. In contrast to the closely related ${\mathrm{GaV}}_{4}{\mathrm{S}}_{8}$, this relaxation dynamics that most likely mirrors coupled orbital and polar fluctuations does not seem to be related to the dynamic processes detected in the THz regime.

Orbital order and fluctuations in the two-leg ladder materials BaFe2X3 ( X=S and Se) and CsFe2Se3

The electronic structure of BaFe$_2X_3$ ($X$ = S and Se) and CsFe$_2$Se$_3$ in which two-leg ladders are formed by the Fe sites are studied by means of x-ray absorption and resonant inelastic x-ray scattering spectroscopy. The x-ray absorption spectra at the Fe L edges for BaFe$_2X_3$ exhibit two components, indicating that itinerant and localized Fe 3$d$ sites coexist. Substantial x-ray linear dichroism (XLD) is observed in polarization dependent spectra, indicating the existence of orbital order or fluctuation in the Fe-ladder even above the N\'{e}el temperature $T_N$. Direct exchange interaction along the legs of the Fe-ladder stabilizes the orbital and antiferromagnetic orders in BaFe$_2$S$_3$, while the ferromagnetic molecular orbitals are realized between the rungs in CsFe$_2$Se$_3$.

Superconductivity without a hole pocket in electron-doped FeSe: Analysis beyond the Migdal-Eliashberg formalism

High-Tc pairing mechanism absent of hole-pockets in heavily electron-doped FeSe is one of the key unsolved problems in Fe-based superconductors. Here, this problem is attacked by focusing on the higher-order many-body effects neglected in conventional Migdal-Eliashberg formalism. We uncover two significant many-body effects for high-Tc superconductivity: (i) Due to the "vertex correction", the dressed multiorbital Coulomb interaction acquires prominent orbital dependence for low-energy electrons. The dressed Coulomb interaction not only induces the orbital fluctuations, but also magnifies the electron-boson coupling constant. Therefore, moderate orbital fluctuations give strong attractive pairing interaction. (ii) The "multi-fluctuation-exchange pairing process" causes large inter-pocket attractive force, which is as important as usual single-fluctuation-exchange process. Due to these two significant effects dropped in the Migdal-Eliashberg formalism, the anisotropic $s_{++}$-wave state in heavily electron-doped FeSe is satisfactorily explained. The proposed "inter-electron-pocket pairing mechanism" will enlarge Tc in other Fe-based superconductors.

Plain s-Wave Superconductivity near Magnetic Criticality: Enhancement of Attractive Electron–Boson Coupling Vertex Corrections

Recent experiments revealed that the plain $s$-wave state without any sign-reversal emerges in various metals near the magnetic criticality. To understand this counter-intuitive phenomenon, we study the gap equation for the multiorbital Hubbard-Holstein model, by analyzing the vertex correction (VC) due to the higher-order electron-correlation effects. We find that the phonon-mediated orbital fluctuations are magnified by the VC for the susceptibility ($\chi$-VC). In addition, the charge-channel attractive interaction is enlarged by the VC for the coupling-constant ($U$-VC), which is significant when the interaction has prominent $q$-dependences so the Migdal theorem fails. Due to both $\chi$-VC and $U$-VC, the plain $s$-wave state is caused by the small electron-phonon interaction near the magnetic criticality against the repulsive Coulomb interaction. We find that the direct Coulomb repulsion for the plain $s$-wave Cooper pair is strongly reduced by the "multiorbital screening effect".

Mechanism of the high transition temperature for the 1111-type iron-based superconductors RFeAsO (R=rareearth) : Synergistic effects of local structures and 4f electrons

Among the iron-based superconductors, the 1111-type Fe-As--based superconductors $\mathrm{REFeAs}{\mathrm{O}}_{1\text{\ensuremath{-}}x}{\mathrm{F}}_{x}$ (RE = rare earth) exhibit high transition temperatures $({T}_{c})$ above 40 K. We perform first-principles calculations based on density functional theory with the consideration of both electronic correlations and spin-orbit couplings on rare earths and Fe ions to study the underlying mechanism as the microscopic structural distortions in REFeAsO tuned by both lanthanide contraction and external strain. The electronic structures evolve similarly in both cases. It is found that there exist an optimal structural regime that will not only initialize but also optimize the orbital fluctuations due to the competing Fe-As and Fe-Fe crystal fields. We also find that the key structural features in REFeAsO, such as As-Fe-As bond angle, intrinsically induce the modification of the Fermi surface and dynamic spin fluctuation. These results suggest that the superconductivity is mediated by antiferromagnetic spin fluctuations. Simultaneously, we show that the rare-earth $4f$ electrons play important roles on the high transition temperature whose behavior might be analogous to that of the heavy-fermion superconductors. The superconductivity of these 1111-type iron-based superconductors with high-${T}_{c}$ is considered to originate from the synergistic effects of local structures and $4f$ electrons.

Three superconducting phases with different categories of pairing in hole- and electron-doped LaFeAs1−xPxO

The phase diagram of LaFeAs$_{1-x}$P$_x$O system has been extensively studied through hole- and electron-doping as well as As/P-substitution. It has been revealed that there are three different superconducting phases with different Fermi surface (FS) topologies and thus with possibly different pairing glues. One of them is well understood as spin fluctuation-mediated superconductivity within a FS nesting scenario. Another one with the FSs in a bad nesting condition must be explained in a different context such as orbital or spin fluctuation in strongly correlated electronic system. In both phases, $T$-linear resistivity was commonly observed when the superconducting transition temperature $T_{\rm c}$ becomes the highest value, indicating that the strength of bosonic fluctuation determines $T_{\rm c}$. In the last superconducting phase, the nesting condition of FSs and the related bosonic fluctuation are moderate. Variety of phase diagram characterizes the multiple orbital nature of the iron-based superconductors which are just near the boundary between weak and strong correlation regimes.

Investigation of the multiplet structures and crystal field effects of a TiO6 3d 1 cluster based on configuration interaction calculations

Configuration interaction cluster calculation can effectively reproduce the experimentally measured Ti L 23-edge absorption spectrum for the TiO6 cluster LaTiO3. A further investigation of the hybridization strength and charge-transfer energy effects on the multiplet structures suggests that LaTiO3 should be classified as an intermediate state between the charge-transfer and Mott–Hubbard regimes. Detailed temperature-dependent simulations of absorption spectra support the lifting of Ti t 2g orbital degeneracy and crystal field splitting. The spin–orbit coupling scenario is ruled out, even though 3d spin–orbit coupling can reproduce the experimental spectrum without including temperature. A combined polarization- and crystal-field-splitting-dependent analysis indicates asymmetric ΔCF–orbital interactions for the TiO6 cluster [Ti3+:3d 1(t 2g 1)], different from the orbital–lattice interactions reported for the NiO6 cluster [Ni3+:3d 7(t 2g 6 eg 1)]. The orbital polarization is defined in terms of the normalized electron occupancies in orbitals with xy and xz(yz) symmetries, and nearly complete orbital polarization (more than 75%) is observed, indicating strongly reduced orbital fluctuations due to the correlation effects. This is consistent with the density of states for titanates based on local density approximation plus dynamical mean-field theory calculations.

Combined First-Principles Calculations and Experimental Study of the Phonon Modes in the Multiferroic Compound GeV4S8

The lattice dynamics of the GeV4S8 compound has been investigated using both density functional calculations and Raman/infrared (IR) measurements. While the accordance between the computed and the experimental data is very good in the low-temperature, ferroelectric phase (25K, Imm2), this is not the case for the high-temperature, paraelectric one within the F43m group. Using group theory and first-principles calculations, we show that the IR/Raman phonon modes are, however, compatible with the I4m2 space group. Analysis of the different modes at the ferroelectric transition shows that simultaneous weakening/strengthening of two symmetry-related bonds within the V4S4 cluster is a direct consequence of the orbital-order instability driving the ferroelectric transition. The softening of modes associated with such distortions call for strong orbital occupation fluctuations above Tc. These fluctuations, associated with the discrepancy between the X-ray scattering F43m group and the lattice dynamics I4m2 gr...

Interplay between Magnetism, Superconductivity, and Orbital Order in 5-Pocket Model for Iron-Based Superconductors: Parquet Renormalization Group Study.

We report the results of the parquet renormalization group (RG) analysis of the phase diagram of the most general 5-pocket model for Fe-based superconductors. We use as an input the orbital structure of excitations near the five pockets made out of d_{xz}, d_{yz}, and d_{xy} orbitals and argue that there are 40 different interactions between low-energy fermions in the orbital basis. All interactions flow under the RG, as one progressively integrates out fermions with higher energies. We find that the low-energy behavior is amazingly simple, despite the large number of interactions. Namely, at low energies the full 5-pocket model effectively reduces either to a 3-pocket model made of one d_{xy} hole pocket and two electron pockets or a 4-pocket model made of two d_{xz}/d_{yz} hole pockets and two electron pockets. The leading instability in the effective 4-pocket model is a spontaneous orbital (nematic) order, followed by s^{+-} superconductivity. In the effective 3-pocket model, orbital fluctuations are weaker, and the system develops either s^{+-} superconductivity or a stripe spin-density wave. In the latter case, nematicity is induced by composite spin fluctuations.

Role of vertex corrections in the matrix formulation of the random phase approximation for the multiorbital Hubbard model

In the framework of a multiorbital Hubbard model description of superconductivity, a matrix formulation of the superconducting pairing interaction that has been widely used is designed to treat spin, charge and orbital fluctuations within a random phase approximation (RPA). In terms of Feynman diagrams, this takes into account particle-hole ladder and bubble contributions as expected. It turns out, however, that this matrix formulation also generates additional terms which have the diagrammatic structure of vertex corrections. Here we examine these terms and discuss the relationship between the matrix-RPA superconducting pairing interaction and the Feynman diagrams that it sums.

Structure of the pairing gap from orbital nematic fluctuations

We study superconducting instability from orbital nematic fluctuations in a minimal model consisting of the $d_{xz}$ and $d_{yz}$ orbitals, and choose model parameters which capture the typical Fermi surface geometry observed in iron-based superconductors. We solve the Eliashberg equations down to low temperatures with keeping the renormalization function and a full momentum dependence of the pairing gap. When superconductivity occurs in the tetragonal phase, we find that the pairing gap exhibits a weak momentum dependence over the Fermi surfaces. The superconducting instability occurs also inside the nematic phase. When the $d_{xz}$ orbital is occupied more than the $d_{yz}$ orbital in the nematic phase, a larger (smaller) gap is realized on the Fermi-surface parts, where the $d_{xz}$ ($d_{yz}$) orbital component is dominant, leading to a substantial momentum dependence of the pairing gap on the hole Fermi surfaces. On the other hand, the momentum dependence of the gap is weak on the electron Fermi surfaces. We also find that while the leading instability is the so-called $s_{++}$-wave symmetry, the second leading one is $d_{x^2-y^2}$-wave symmetry. In particular, these two states are nearly degenerate in the tetragonal phase whereas such quasi-degeneracy is lifted in the nematic phase and the $d_{x^2-y^2}$-wave symmetry changes to highly anisotropic $s$-wave symmetry.

The role of spin–orbit coupling in the electronic structure of iron‐based superconductors

Spin–orbit coupling (SOC) is a fundamental interaction in solids that can induce a broad spectrum of unusual physical properties from topologically non‐trivial insulating states to unconventional pairing in superconductors. In iron‐based superconductors (IBS) its role has so far been considered insignificant with the models based on spin‐ or orbital fluctuations pairing being the most advanced in the field. In this review we discuss the recent direct observation of SOC by angle‐resolved photoemission spectroscopy (ARPES).Electron‐like Fermi surface of LiFeAs split by SOC.

Enhancing superconductivity inA3C60fullerides

Motivated by the recent experimental report of a possible light-induced superconductivity in A3C60 at high temperature [Mitrano et al., Nature 530, 451 (2016)], we investigate theoretical mechanisms for enhanced superconductivity in A3C60 fullerenes. We find that an `interaction imbalance' corresponding to a smaller value of the Coulomb matrix element for two of the molecular orbitals in comparison to the third one, efficiently enhances superconductivity. Furthermore, we perform first-principle calculations of the changes in the electronic structure and in the screened Coulomb matrix elements of A3C60, brought in by the deformation associated with the pumped T1u intra-molecular mode. We find that an interaction imbalance is indeed induced, with a favorable sign and magnitude for superconductivity enhancement. The physical mechanism responsible for this enhancement consists in a stabilisation of the intra-molecular states containing a singlet pair, while preserving the orbital fluctuations allowing for a coherent inter-orbital delocalization of the pair. Other perturbations have also been considered and found to be detrimental to superconductivity. The light-induced deformation and ensuing interaction imbalance is shown to bring superconductivity further into the strong-coupling regime.

Exotic Spin-Orbital Physics in Hybrid Oxides

We compare the effective spin-orbital superexchange triggered by magnetic 3d impurities with d3 and d2 configurations and either no orbital degree of freedom (orbital dilution) or hole replacing a doublon (charge dilution) in a 4d4 Mott insulator with S=1 spins. Impurities causing orbital dilution act either as spin defects decoupled from the surrounding ions or generate orbital polarons along d3- d4 hybrid bonds. The exchange on these bonds determines which orbital is occupied by a doublon on the host site. In case of charge dilution by 3d2 impurities additional propto T_{i}^{+}T_{j}^{+} terms arise which enhance orbital fluctuations. We show that such terms may radically change orbital pattern at relatively low doping by x=1/8 hole defects. Our findings provide new perspective for future theoretical and experimental studies of doped transition metal oxides.

Functional renormalization group study of orbital fluctuation mediated superconductivity: Impact of the electron-boson coupling vertex corrections

In various multiorbital systems, the emergence of the orbital fluctuations and its role on the pairing mechanism attract increasing attention. To archive deep understanding on these issues, we perform the functional-renormalization-group (fRG) study for the two-orbital Hubbard model. The vertex corrections for the electron-boson coupling ($U$-VC), which are dropped in the Migdal-Eliashberg gap equation, are obtained by solving the RG equation. We reveal that the dressed electron-boson coupling for the charge-channel, $U_{eff}^c$, becomes much larger than the bare Coulomb interaction, $U^0$, due to the $U$-VC in the presence of moderate spin fluctuations. For this reason, the attractive pairing interaction due to the charge or orbital fluctuations is enlarged by the factor $(U_{eff}^c/U^0)^2 >> 1$. In contrast, the spin fluctuation pairing interaction is suppressed by the spin-channel $U$-VC, because of the relation $U_{eff}^s << U^0$. The present study demonstrates that the orbital or charge fluctuation pairing mechanism can be realized in various multiorbital systems thanks to the $U$-VC, such as in Fe-based superconductors.