Reduced dimensionality and strong electronic correlations, which are among the most important ingredients for cupratelike high-Tc superconductivity, characterize also the physics of nickelate-based heterostructures. Starting from the local-density approximation we arrive at a simple two-band model for quasi-two-dimensional 2D LaNiO3 /LaAlO3 heterostructures and extend it by introducing an appropriate hopping in the z direction to describe the dimensional crossover to three dimensions 3D. Using dynamical mean-field theory, we study the effects of electronic correlations with increasing interaction strength along the crossover from 2D to 3D. Qualitatively, the effects of electronic correlations are surprisingly similar, albeit quantitatively larger interaction strengths are required in three dimensions for getting a Mott-Hubbard insulating state. The exchange parameters of an effective Kugel-Khomskii-type spin-orbital model are also derived and reveal strong antiferromagnetic tendencies.
Layer-by-layer oxide molecular beam epitaxy has been used to synthesize cuprate-nickelate multilayer structures of composition (La$_2$CuO$_4$)$_m$/LaO/(LaNiO$_3$)$_n$. In a combined experimental and theoretical study, we show that these structures allow a clean separation of dopant and doped layers. Specifically, the LaO layer separating cuprate and nickelate blocks provides an additional charge that, according to density functional theory calculations, is predominantly accommodated in the interfacial nickelate layers. This is reflected in an elongation of bond distances and changes in valence…
Resonant inelastic x-ray scattering is used to investigate the electronic origin of orbital polarization in nickelate heterostructures taking $\mathrm{LaTiO_3-LaNiO_3-3x(LaAlO_3)}$, a system with exceptionally large polarization, as a model system. We find that heterostructuring generates only minor changes in the Ni $3d$ orbital energy levels, contradicting the often-invoked picture in which changes in orbital energy levels generate orbital polarization. Instead, O $K$-edge x-ray absorption spectroscopy demonstrates that orbital polarization is caused by an anisotropic reconstruction of the o…
The electronic structure of transition metal oxides featuring correlated electrons can be rationalized within the Zaanen-Sawatzky-Allen framework. Following a brief description of the present paradigms of electronic behavior, we focus on the physics of rare earth nickelates as an archetype of complexity emerging within the charge transfer regime. The intriguing prospect of realizing the physics of high $T_c$ cuprates through heterostructuring resulted in a massive endeavor to epitaxially stabilize these materials in ultra-thin form. A plethora of new phenomena unfolded in such artificial struc…
The prediction of high $T_c$ superconductivity in layers of LaNiO$_3$ through orbital engineering has led to extensive research efforts over the last fifteen years. During this period, a plethora of thin films and heterostructures based rare-earth nickelate family with perovskite structure has been synthesized and explored. In this short perspective, we briefly review the complexity of bulk $RE$NiO$_3$, spotlighting several recent findings of emergent phenomena in heterostructures containing the interface between $RE$NiO$_3$ and another transition metal oxide. Finally, we outline potentially i…
The metal-insulator transition (MIT) of bulk rare-earth nickelates is accompanied by a simultaneous charge ordering (CO) transition. We have investigated low-frequency resistance fluctuations (noise) across the MIT and magnetic transition of [EuNiO$_3$/LaNiO$_3$] superlattices, where selective suppression of charge ordering has been achieved by mismatching the superlattice periodicity with the periodicity of charge ordering. We have observed that irrespective of the presence/absence of long-range CO, the noise magnitude is enhanced by several orders with strong non-1/$f$ ($f$ = frequency) comp…
Heterostructure engineering provides an efficient way to obtain several unconventional phases of LaNiO3, which is otherwise paramagnetic, metallic in bulk form. In this work, a new class of short periodic superlattices, consisting of LaNiO3 and EuNiO3 have been grown by pulsed laser interval deposition to investigate the effect of structural symmetry mismatch on the electronic and magnetic behaviors. Synchrotron based soft and hard X-ray resonant scattering experiments have found that these heterostructures undergo simultaneous electronic and magnetic transitions. Most importantly, LaNiO3 with…
We have used resonant x-ray diffraction to develop a detailed description of antiferromagnetic ordering in epitaxial superlattices based on two-unit-cell thick layers of the strongly correlated metal LaNiO3. We also report reference experiments on thin films of PrNiO3 and NdNiO3. The resulting data indicate a spiral state whose polarization plane can be controlled by adjusting the Ni d-orbital occupation via two independent mechanisms: epitaxial strain and quantum confinement of the valence electrons. The data are discussed in the light of recent theoretical predictions.
We present a combined study of Ni $K$-edge resonant x-ray scattering and density functional calculations to probe and distinguish electronically driven ordering and lattice distortions in nickelate heterostructures. We demonstrate that due to the low crystal symmetry, contributions from structural distortions can contribute significantly to the energy-dependent Bragg peak intensities of a bond-ordered NdNiO$_3$ reference film. For a LaNiO$_3$-LaAlO$_3$ superlattice that exhibits magnetic order, we establish a rigorous upper bound on the bond-order parameter. We thus conclusively confirm predic…
Using the local density approximation and its combination with dynamical mean-field theory, we show that electronic correlations induce a single-sheet, cuprate-like Fermi surface for hole-doped 1/1 LaNiO3 /LaAlO3 heterostructures, even though both eg orbitals contribute to it. The Ni 3d 3z^2-1 orbital plays the role of the axial Cu 4s-like orbital in the cuprates. These two results indicate that "orbital engineering" by means of heterostructuring should be possible. As we also find strong antiferromagnetic correlations, the low-energy electronic and spin excitations in nickelate heterostructur…
Transition metal based oxide heterostructures exhibit diverse emergent phenomena e.g. two dimensional electron gas, superconductivity, non-collinear magnetic phase, ferroelectricity, polar vortices, topological Hall effect etc., which are absent in the constituent bulk oxides. The microscopic understandings of these properties in such nanometer thick materials are extremely challenging. Synchrotron x-ray based techniques such as x-ray diffraction, x-ray absorption spectroscopy (XAS), resonant x-ray scattering (RXS), resonant inelastic x-ray scattering (RIXS), x-ray photoemission spectroscopy, …
Bulk magnetic order in two dimensional La4Ni3O8 nickelate with Ni1+/Ni2+ (d9/d8), isoelectronic with superconducting cuprates is demonstrated experimentally and theoretically. Magnetization, specific heat and 139La NMR evidence a transition at 105 K to an antiferromagnetic state. Theoretical calculations by DFT relate the transition to a nesting instability of the Fermi surface with ordering wave-vector Q = [1/3, 1/3, 0].
The discovery of superconductivity in infinite-layer nickelates has attracted much attention due to their association to the high-$T_c$ cuprates. Cuprate superconductivity was first demonstrated in bulk samples and subsequently in thin films. In the nickelates, however, the situation has been reversed: although surging as a bulk phenomenon, nickelate superconductivity has only been reported in thin films so far. At the same time, the specifics of infinite-layer nickelates yield distinct interface and surface effects that determine their bulk vs thin-film behavior. In this paper, we provide an …
Unveiling orbital landscapes in strongly correlated bulk nickelates with $s$-NIXS
Max Planck Institute for Chemical Physics of Solids（马克斯·普朗克固体化学物理研究所）；Deutsches Elektronen-Synchrotron (DESY)（德国电子同步辐射中心）；Max Planck Institute for Solid State Research（马克斯·普朗克固相研究所）
Perovskite nickelate heterostructure consisting of single unit cell of EuNiO$_3$ and LaNiO$_3$ have been grown on a set of single crystalline substrates by pulsed laser interval deposition to investigate the effect of epitaxial strain on electronic and magnetic properties at the extreme interface limit. Despite the variation of substrate in-plane lattice constants and lattice symmetry, the structural response to heterostructuring is primarily controlled by the presence of EuNiO$_3$ layer. In sharp contrast to bulk LaNiO$_3$ or EuNiO$_3$, the superlattices grown under tensile strains exhibit me…
We analyze the electronic structure of different surface terminations for infinite-layer nickelates. Surface NiO$_2$ layers are found to be buckled, in contrast to planar bulk layers. While the rare-earth terminated surface fermiology is similar to the bulk limit of the nickelates, the NiO$_2$ terminated surface band structure is significantly altered, originating from the effect of absence of rare-earth atoms on the crystal field splitting. Contrary to the bulk Fermi surfaces, there are two Ni-$3d$ Fermi pockets, giving rise to enhanced spectral weight around the $\bar{\text{M}}$ point in the…
We present a general methodology that enables the disentanglement of the electronic and lattice contributions to the metal-insulator transition by building an energy landscape from numerical solutions of the equation of state. The methodology works with any electronic structure method that provides electronic expectation values at given atomic positions. Applying the theory to rare-earth perovskite nickelates ($R$NiO$_3$) and Ruddlesden-Popper calcium ruthenates (Ca$_2$RuO$_4$) in bulk, heterostructure and epitaxially strained thin film forms using equation of state results from density functi…
Bulk superconductivity in quintuple-layer square-planar nickelate La6Ni5O12
University of Science and Technology of China（中国科学技术大学）；Collaborative Innovation Center of Advanced Microstructures, Nanjing University（南京大学先进微结构协同创新中心）
Infinite-layer (IL) nickelates are an emerging class of superconductors, where the Ni$^{1+}$ valence state in a square planar NiO$_2$ coordination can only be reached via topotactic reduction of the perovskite phase. However, this topotactic soft chemistry with hydrogenous reagents is still at a stage of rapid development, and there is a number of open issues, especially considering the possibility of hydrogen incorporation. Here we study the time dependence of the topotactic transformation of LaNiO$_3$ to LaNiO$_2$ for powder samples with x-ray diffraction and gas extraction techniques. While…
We report a density functional theory plus dynamical mean field theory (DFT+DMFT) study of an oxide heterostructure of LaNiO$_3$ (LNO) bilayers in [111] direction interleaved with four atomic monolayers of LaAlO$_3$. DFT+$U$ optimizations yield two stable solutions: a uniform structure with equivalent NiO$_6$ octahedra, as well as a bond-disproportionated (BD) structure featuring a breathing distortion. For both structures, we construct the low-energy models describing the Ni $e_g$ states by means of Wannier projections supplemented by the Kanamori interaction, and solve them by DMFT. Using th…
We used high-resolution resonant inelastic x-ray scattering (RIXS) at the Ni $L_3$ edge to simultaneously investigate high-energy interband transitions characteristic of Ni-O bond ordering and low-energy collective excitations of the Ni spins in the rare-earth nickelates $R$NiO$_3$ ($R$ = Nd, Pr, La) with pseudocubic perovskite structure. With the support of calculations based on a double-cluster model we quantify bond order (BO) amplitudes for different thin films and heterostructures and discriminate short-range BO fluctuations from long-range static order. Moreover we investigate magnetic o…
Bulk superconductivity up to 96 K in pressurized nickelate single crystals
State Key Laboratory of Crystal Materials, Institute of Crystal Materials, Shandong University（山东大学晶体材料研究所）；Center for High Pressure Science and Technology Advanced Research（高压科学和技术先进研究中心）；Shanghai Key Laboratory of Material Frontiers Research in Extreme Environments (MFree), Shanghai Advanced Research in Physical Sciences (SHARPS)（极端环境材料前沿研究上海市重点实验室（MFree），上海物理科学先进研究中心（SHARPS））；Beijing National Laboratory for Condensed Matter Physics and Institute of Physics, Chinese Academy of Sciences（中国科学院物理研究所，凝聚态物理国家实验室）；School of Physical Sciences, University of Chinese Academy of Sciences（中国科学院大学物理科学学院）；CAS Key Laboratory of Standardization and Measurement for Nanotechnology, National Center for Nanoscience and Technology（中国科学院纳米技术标准化与计量重点实验室，国家纳米科学中心）；Key Laboratory of Materials Physics, Ministry of Education, School of Physics, Zhengzhou University（郑州大学物理学院，教育部材料物理重点实验室）；Institute of Quantum Materials and Physics, Henan Academy of Sciences（河南省科学院量子材料与物理研究所）；NSF’s ChemMatCARS, The University of Chicago（芝加哥大学，美国国家科学基金会化学-材料束线组合站（ChemMatCARS））；School of Materials and Energy, University of Electronic Science and Technology of China（电子科技大学材料与能源学院）；Huzhou Key Laboratory of Smart and Clean Energy, Yangtze Delta Region Institute (Huzhou), University of Electronic Science and Technology of China（电子科技大学长三角研究院（湖州），湖州市智能清洁能源重点实验室）；Shanghai Synchrotron Radiation Facility, Shanghai Advanced Research Institute, Chinese Academy of Sciences（中国科学院上海高等研究院，上海同步辐射光源）；Japan Synchrotron Radiation Research Institute（日本同步辐射研究机构）；NanoTerasu Promotion Division, Japan Synchrotron Radiation Research Institute (JASRI)（日本同步辐射研究机构（JASRI）纳米terasu推进部）
Motivated by recent Fermi surface and transport measurements on LaNiO3, we study the Mott Metal-Insulator transitions of perovskite nickelates, with the chemical formula RNiO3, where R is a rare-earth ion. We introduce and study a minimal two-band model, which takes into account only the eg bands. In the weak to intermediate correlation limit, a Hartree-Fock analysis predicts charge and spin order consistent with experiments on R=Pr, Nd, driven by Fermi surface nesting. It also produces an interesting semi-metallic electronic state in the model when an ideal cubic structure is assumed. We also…
$d$-wave superconductivity in quintuple-layer bulk superconductor La$_6$Ni$_5$O$_{12}$
Beijing National Laboratory for Condensed Matter Physics（北京凝聚态物理国家研究中心）；Institute of Physics, Chinese Academy of Sciences（中国科学院物理研究所）；Kavli Institute for Theoretical Sciences, University of Chinese Academy of Sciences（中国科学院大学卡尔丹理论科学研究所）；School of Physical Science and Technology, ShanghaiTech University（上海科技大学理学院）；School of Physical Sciences, University of Chinese Academy of Sciences（中国科学院大学物理科学学院）；New Cornerstone Science Laboratory（新Cornerstone科学实验室）
Rare-earth nickelates RNiO$_2$ adopting an infinite-layer phase show superconductivity once La, Pr or Nd are substituted by a divalent cation. Either in the pristine or doped form, these materials are reported to adopt a high symmetry, perfectly symmetric, P4/mmm tetragonal cell. Nevertheless, bulk compounds are scarce, hindering a full understanding of the role of chemical pressure or strain on lattice distortions that in turn could alter magnetic and electronic properties of the 2D nickelates. Here, by performing a full analysis of the prototypical YNiO$_2$ compound with first-principles sim…
The origin of simultaneous electronic, structural and magnetic transitions in bulk rare-earth nickelates ($RE$NiO$_3$) remains puzzling with multiple conflicting reports on the nature of these entangled phase transitions. Heterostructure engineering of these materials offers unique opportunity to decouple metal-insulator transition (MIT) from the magnetic transition. However, the evolution of underlying electronic properties across these decoupled transitions remains largely unexplored. In order to address this, we have measured Hall effect on a series of epitaxial NdNiO$_3$ films, spanning a …
Quantum ground states on the non-trivial side of a topological quantum critical point (TQCP) have unique properties that make them attractive candidates for quantum information applications. A recent example is provided by s-wave superconductivity on a semiconductor platform, which is tuned through a TQCP to a topological superconducting (TS) state by an external Zeeman field. Despite many attractive features of TS states, TQCPs themselves do not break any symmetries, making it impossible to distinguish the TS state from a regular superconductor in conventional bulk measurements. Here we show …
The infinite-layer structure nickelate Ba$_2$NiO$_2$(AgSe)$_2$ (BNOAS) with $d^8$ Ni ions and a peculiar susceptibility $χ(T)$ is studied with correlated density functional methods. The overriding feature of the calculations is violation of Hund's rule coupled with complete but unconventional spin-orbital polarization, leading to an unexpected low spin $^1B_1$, "off-diagonal singlet" (ODS) textured by an internal orbital structure of compensating $d_{x^2-y^2}^{\uparrow}$ and $d_{z^2}^{\downarrow}$ spins. This unconventional configuration has lower energy than conventional high-spin or low-spin…
We consider heterostructures obtained by stacking layers of two s-wave superconductors with significantly different coupling strengths, respectively in the weak- and strong-coupling regimes. The weak- and strong-coupling superconductors are chosen with similar critical temperatures for bulk systems. Using dynamical mean-field theory methods, we find an ubiquitous enhancement of the superconducting critical temperature for all the heterostructures where a single layer of one of the two superconductors is alternated with a thicker multilayer of the other. Two distinct physical regimes can be ide…
We perform core-level X-ray photoemission spectroscopy (XPS) and electronic Raman scattering studies of electronic structures and spin fluctuations in the bulk samples of the nickelate oxide NdNiO$_2$. According to Nd $3d$ and O $1s$ XPS spectra, we conclude that NdNiO$_2$ has a large transfer energy. From the analysis of the main line of the Ni $2p_{3/2}$ XPS, we confirm the NiO$_2$ planes in NdNiO$_2$ are of Mott-Hubbard type in the Zaanen-Sawatzky-Allen scheme. The two-magnon peak in the Raman scattering provides direct evidence for the strong spin-fluctuation in NdNiO$_2$. The peak positio…
Thermodynamic phase transition, pairing symmetry and Fermi surface topology in Ruddlesden-Popper nickelate films
University of Science and Technology of China（中国科学技术大学）；Hefei National Laboratory, University of Science and Technology of China（合肥国家实验室，中国科学技术大学）；Hefei National Research Center for Physical Sciences at the Microscale, University of Science and Technology of China（中国科学院微观结构国家重点实验室，中国科学技术大学）
Fractional topological insulators (FTI) are electronic topological phases in $(3+1)$ dimensions enriched by time reversal (TR) and charge $U(1)$ conservation symmetries. We focus on the simplest series of fermionic FTI, whose bulk quasiparticles consist of deconfined partons that carry fractional electric charges in integral units of $e^\ast=e/(2n+1)$ and couple to a discrete $\mathbb{Z}_{2n+1}$ gauge theory. We propose massive symmetry preserving or breaking FTI surface states. Combining the long-ranged entangled bulk with these topological surface states, we deduce the novel topological orde…
Ultrafast Magneto-Pressure Spectroscopy and Control of Correlated Phases in a Trilayer Nickelate
Ames National Laboratory, U.S. Department of Energy（艾姆斯国家实验室）；Department of Physics and Astronomy, Iowa State University（爱荷华州立大学物理与天文学系）；Materials Science Division, Argonne National Laboratory（阿贡国家实验室材料科学部）；School of Physics, Southeast University（东南大学物理学院）；Department of Physics, University of Alabama at Birmingham（阿拉巴马大学伯明翰分校物理系）
We develop the multi-scale ab-initio scheme for correlated electrons (MACE) for transitionmetal-oxide heterostructures, and determine the parameters of the low-energy effective model. By separating Ti t2g bands near the Fermi level from the global Kohn-Sham (KS) bands of LaAlO3/SrTiO3 which are highly entangled with each other, we are able to calculate the parameters of the low-energy effective model of the interface with the help of constrained random phase approximation (cRPA). The on-site energies of the Ti t2g orbitals in the 1stlayer is about 650 meV lower than those in the 2nd-layer. In …
Enhanced $s^\pm$-wave superconductivity in electron-doped La$_3$Ni$_2$O$_7$
Institute for Quantum Science, School of Physical Science and Technology, Soochow University（苏州大学物理科学与技术学院量子科学研究所）；School of Physics, Northwest University（西北大学物理学院）；Institute of Solid State Physics, TU Wien（维也纳工业大学固体物理研究所）；Shaanxi Key Laboratory for Theoretical Physics Frontiers（陕西省理论物理前沿重点实验室）；Fundamental Discipline Research Center for Quantum Science and technology of Shaanxi Province（陕西省量子科学与技术基础学科研究中心）；State Key Laboratory of Surface Physics and Department of Physics, Fudan University（复旦大学表面物理国家重点实验室及物理系）
In this work we present a model system built out of artificially layered materials, allowing us to understand the interrelation of magnetic phases with that of the metallic-insulating phase at long length-scales, and enabling new strategies for the design and control of materials in devices. The artificial model system consists of superlattices made of SmNiO$_3$ and NdNiO$_3$ layers -- two members of the fascinating rare earth nickelate family, having different metal-to-insulator and magnetic transition temperatures. By combining two complementary techniques -- resonant elastic x-ray scatterin…
Nearly perfect Fermi surface nesting in hole-doped La$_3$Ni$_2$O$_7$ enables bulk superconductivity without pressure or strain
NYU-ECNU Institute of Physics, NYU Shanghai（纽约大学-华东师范大学物理研究所，上海纽约大学）；Key Laboratory of Polar Materials and Devices, Ministry of Education, East China Normal University（华东师范大学极化材料与器件教育部重点实验室）；Department of Physics, Brown University（布朗大学物理系）；Department of Physics, New York University（纽约大学物理系）
In this paper, we investigate the ground state properties of the $\rm{RMnO}_3/\rm{AMnO}_3$ ($\rm{RMO/AMO}$) heterostructures (R=trivalent cation, and A=divalent cation) by using a two-orbital double-exchange model supplemented by the Poisson's equation. We find that the state stabilized near the interface of the heterostructure is similar to the state of the bulk compound $\rm{(R,A)MO}$ at electronic density close to 0.5. Depending on the bandwidth it will be a charge and orbital ordered CE state or an A-AF state with $x^2-y^2$ orbital order. Our results can explain some properties of long-per…
Strong Coulomb repulsion and spin-orbit coupling are known to give rise to exotic physical phenomena in transition metal oxides. Initial attempts to investigate systems where both of these fundamental interactions are comparably strong, such as 3d and 5d complex oxide superlattices, have revealed properties that only slightly differ from the bulk ones of the constituent materials. Here, we observe that the interfacial coupling between the 3d antiferromagnetic insulator SrMnO3 and the 5d paramagnetic metal SrIrO3 is enormously strong, yielding an anomalous Hall response as the result of charge …
Topotactic transformations between related crystal structures are a powerful emerging route for the synthesis of novel quantum materials. Whereas most such "soft chemistry" experiments have been carried out on polycrystalline powders or thin films, the topotactic modification of single crystals -- the gold standard for physical property measurements on quantum materials -- has proven remarkably difficult. Here, we report the topotactic reduction of La$_{1-x}$Ca$_{x}$NiO$_3$ single-crystals to La$_{1-x}$Ca$_{x}$NiO$_{2+δ}$ using CaH$_2$ as the reducing agent. The transformation from the three-d…
Obtaining strong magnetoelectric couplings in bulk materials and heterostructures is an ongoing challenge. We demonstrate that manganite heterostructures of the form ${\rm (Insulator)/(LaMnO_3)_n/(CaMnO_3)_n/(Insulator)}$ show strong multiferroicity in magnetic manganites where ferroelectric polarization is realized by charges leaking from ${\rm LaMnO_3}$ to ${\rm CaMnO_3}$ due to repulsion. Here, an effective nearest-neighbor electron-electron (electron-hole) repulsion (attraction) is generated by cooperative electron-phonon interaction. Double exchange, when a particle virtually hops to its …
Structural symmetry effects on the competition of density waves and superconductivity in bilayer nickelates
Ruhr-Universität Bochum（波鸿鲁尔大学）；Beijing National Laboratory for Condensed Matter Physics and Institute of Physics, Chinese Academy of Sciences（中国科学院物理研究所（北京凝聚态物理国家研究中心））；School of Physical Sciences, University of Chinese Academy of Sciences（中国科学院大学物理科学学院）
A slave rotor-Hartree Fock formalism is presented for studying the properties of the p-d model describing perovskite transition metal oxides, and a flexible and efficient numerical formalism is developed for its solution. The methodology is shown to yield, within an unified formulation, the significant aspects of the rare earth nickelate phase diagram, including the paramagnetic metal state observed for the LaNiO3 and the correct ground-state magnetic order of insulating compounds. It is then used to elucidate ground state changes occurring as morphology is varied from bulk to strained and un-…
Nickel-based complex oxides have served as a playground for decades in the quest for a copper-oxide analog of the high-temperature superconductivity. They may provide clues towards understanding the mechanism and an alternative route for high-temperature superconductors. The recent discovery of superconductivity in the infinite-layer nickelate thin films has fulfilled this pursuit. However, material synthesis remains challenging, direct demonstration of perfect diamagnetism is still missing, and understanding of the role of the interface and bulk to the superconducting properties is still lack…
Obtaining strong magnetoelectric couplings in bulk materials and heterostructures is an ongoing challenge. We demonstrate that manganite heterostructures of the form ${\rm (Insulator)/(LaMnO_3)_n/(CaMnO_3)_n/(Insulator)}$ show strong multiferroicity in magnetic manganites where ferroelectric polarization is realized by charges leaking from ${\rm LaMnO_3}$ to ${\rm CaMnO_3}$ due to repulsion. Here, an effective nearest-neighbor electron-electron (electron-hole) repulsion (attraction) is generated by cooperative electron-phonon interaction. Double exchange, when a particle virtually hops to its …
