aa r X i v : . [ c ond - m a t . s t r- e l ] A p r Strongly Correlated Electrons in Solids
Henri Alloul ∗ LPS - CNRS/Universite Paris Sud, Orsay, France
Abstract
Most emergent properties of the materials discovered since the 1980s are related to the existenceof electron-electron interactions which are large with respect to the kinetic energies and could notbe thoroughly studied before. The occurrence of metal insulator transitions, of exotic magneticand/or superconducting properties in many new compounds have stimulated a large series of ex-perimental and theoretical developments to grasp their physical significance. We present here asimple introduction to the elementary aspects of the physics of electron-electron interactions, whichcould be a starting point for typical undergraduate students.
The study of the electronic properties of solids, done within an independent electron approximationsince World War II, has been essential for the understanding of the occurrence of semi-conductors.This understanding was at the origin of the information technologies which expanded rapidly afterthe war. But during that period, a myriad of new materials with increasing complexity have beendiscovered as well. These materials were found to display unexpected novel electronic properties. Manysuch properties are not explained by the independent electron approximations, require new conceptualdevelopments, and will certainly lead in the future to specific promising applications. Most of theseemergent properties are linked with magnetic responses due to the strong electron-electron interactionsin these complex new materials.We will briefly discuss how these electronic interactions yield original states of electronic matter.A variety of experimental and theoretical techniques have been developed which permit a detailedinvestigation of their unexpected properties.This article will be organised as follows. Electronic properties of solids were, in the first half of thetwentieth century, considered mostly in the frame of an independent electron approximation with spindegeneracy. The resulting electronic band structure of metals which will be briefly recalled in section 2is such that each electronic level could be doubly occupied. In such an approach one expects metals orinsulators with no significant magnetic properties.In order to explain that some solids display magnetic properties one must reassess the underlyingapproximations that led to the band theory, and especially the averaging approach to the Coulombinteractions between electrons. In section 3 we shall show that one has to take into account thestrong local coulomb repulsion on atomic orbitals, which permits magnetic atomic states and magneticinsulators in the solid state.We shall then specifically mention in section 4 the superconducting state which is an original corre-lated electronic state which occurs in most metals at low temperature. This is a macroscopic quantumelectronic state which results from an indirect electron-electron attractive interaction induced by theinterplay of electronic and atomic vibrations in classical metals in which the electron states do notinteract at high temperatures.We shall then consider in section 5 how electronic correlations yield materials with properties whichare in an intermediate regime between independent, delocalised electrons and local states. Those ∗ Henri Alloul (2014),
Strongly Correlated Electrons in Solids , Scholarpedia, 9(7):32067.
N e and
N a . Here the calculated energy bands are plotted versus a fictitiousdistance between atomic orbitals and r represents the equilibrium distance at ambient temperature.Ne only displays filled or empty bands and is an insulator, while Na has its higher energy band onlyfilled with one electron per atom and is a metal.intermediate electronic states are at the basis of the correlated electron physics. They often displayexotic superconducting states with unexpectedly high transition temperatures, can undergo chargeordering or metal insulator transitions as well as exotic magnetic states considered as spin liquids. Suchoriginal states, which are far from being fully understood at this time, will be introduced in dedicatedScholarpedia articles. Isolated atoms display discrete, narrow electronic levels. In the solid state, the electron can delocalisebetween sites due to the overlap of the electronic orbitals of neighboring atoms. The transfer integrals t between orbitals of neighboring atoms lead to a broadening of the atomic levels into electronic bandswhich characterize the actual band structure of a given material. The width of these energy bands istypically determined by zt where z is the number of neighboring atoms surrounding a given site. Insuch an independent electron approach the available electrons in the material fill the energy levels inincreasing energy order. This yields insulators when filled and empty bands are separated by finitegaps, and metals if there are partially filled energy bands up to an energy level which defines the Fermienergy, as shown in Fig.1. In such an approach, solids with an odd number of electrons per unit cell areexpected to be metals as they should necessarily display partially filled bands in which delocalisationof electrons can be done at moderate energy cost. One distinguishes then among the insulators thecases where the energy gap is small compared to the thermal energy k B T . In that case electrons canbe excited thermally at temperature ∼ T into the first empty band (conduction band) and leave emptyholes in the last occupied band (the valence band), this being the case of semiconductors. Among those,graphene has been highlighted recently, as in that case the gap vanishes and the conduction and valenceband touch each other at a single energy point, the Dirac point which corresponds in that case to theFermi energy.In those cases the band theory for the electronic states applies rather well and explains most ofthe electronic properties of these metals, insulators, semiconductors or Dirac point metals. In all thosecases the independent electrons approach yields a weak paramagnetism as all these descriptions do notlift the spin degeneracy of the electronic states. This Band theory describes these materials well as the2igure 2: The Mott–Hubbard model. (a) The atomic orbitals on atomic sites at a distance d, with thetransfer integrals t between neighboring sites . (b) The levels for the Hubbard model as a function oft or d. On the left: Isolated atom with energy levels ǫ and ǫ + U . Center: Mott–Hubbard insulatorobtained for a small hopping integral W < U . Right: Metallic situation corresponding to
W > U . Thisvery simple approach goes by the name of the Hubbard model.k space construction lifts the site degeneracy of the atomic state by building Bloch states which havedifferent energies and well defined properties under translation. If t is small, then one expects very narrow bands and localized electronic states, as the case t = 0corresponds to strictly isolated atomic states. In that case electronic interactions can no longer betreated as an average as done in band theories and do give rise to local moment and magnetism, as weshall see hereafter. Let us begin by considering the case of an isolated atom (on the left in Fig2). In this context, in bandtheory, it is assumed that the energy brought to the system by an extra electron would be ǫ , and thata second electron on the same atom would also bring ǫ , so that the total energy would be 2 ǫ for adoubly negatively charged ion. But this is obviously not very realistic, owing to Coulomb repulsion.Apart from its ’orbital’ energy ǫ , the second electron will also be subject to the Coulomb repulsion ofthe first electron, and its energy will thus be higher than ǫ by an amount usually denoted by U, whichrepresents the Coulomb repulsion between the first and second electrons added to an initially neutralatom. The total energy of the doubly negative ion is thus ǫ + U . Note that U can vary considerablydepending on the atom (from about 1 eV to more than 10 eV).If we now consider this ion in a crystal, the hopping integrals between nearest neighbors will broadenthe discrete atomic levels into bands of width W = zt . To begin, we consider the limiting case of smallhopping integral compared with U . We find ourselves in a situation corresponding to the middle ofFig.2. There are two allowed energy bands called the upper and lower Hubbard bands, separated by aband gap. This gives the impression that we have a typical insulator (or semiconductor). But this isnot in fact correct. There is one additional one-electron state per atom, so that, in a solid comprising N n atoms, the lower band of the middle column can contain up to N n electrons, rather than up to3 N n electrons, as is the case in the context of the independent electron band theory. In particular, ifthere is now one electron per atom (or more generally an odd number of electrons per primitive cell),the lower band will be completely filled and the upper band completely empty. We will thus have aninsulator with an odd number of electrons per primitive cell, as a consequence of the interactions U between electrons. The very existence of such an insulator (usually called a Mott-Hubbard insulator inrecognition of the two British scientists who first studied them in the 1960s) is thus a consequence ofthe Coulomb interaction between electrons. As we shall see later, important examples of Mott-Hubbardinsulators are undoped cuprates in which the Cu ions are in a 3 d state. While usual band insulators should be nonmagnetic (or more precisely, slightly diamagnetic), verydifferent expectations occur for a Mott-Hubbard insulator. If we begin by considering the limiting caseof very small hopping integrals, we end up with isolated atoms. The electron in the level ǫ can thenhave spin up or spin down, behaving like an isolated spin 1 /
2. In the solid, these spins taken togetherwill give rise to Curie paramagnetism with a spin susceptibility ∼ /T , that is, a paramagnetic insulatorsusceptibility that contradicts band theory. If one takes into account the finite value of the hoppingintegral t , it can be shown that, at low enough temperatures, the spins on neighboring atomic sites willlike to arrange themselves in opposite directions, that is, antiferromagnetic coupling dominates.The main conclusion which can be taken here is that going beyond the possibilities offered byband theory (paramagnetic metals and diamagnetic insulators), the presence of Coulomb interactionsbetween electrons, if they are strong enough, can give rise to an insulating state with a variety ofmagnetic properties, such as Curie paramagnetism, antiferromagnetism (but also ferromagnetism), andso on as will be shown later on.The Hubbard model, which replaces the true Coulomb potential V ( r ) ∼ /r by a repulsion whichonly acts if the two electrons are located on the same atom, is clearly a drastic simplification of the actualphysical situation. However, it is rather naturally justified in the context of the theory of magneticphenomena. Experiments show that there are not only magnetic insulators of spin 1 / s levels (hence non-degenerate), but d - or f -type (hence five- or seven-fold degenerate).In such a situation one has to take in more detail the local repulsive Coulomb interaction betweenelectrons on the orbitals of such poly-electronic atoms. Though the Coulomb interaction is purely elec-trostatic, it differentiates the energy levels of the atomic orbitals, depending of their orbital symmetryand disfavors then double occupancy of some of them. These electronic interactions when combinedwith the Pauli principle are responsible for the local moment magnetism of isolated atoms. In thissituation, the angular momentum of each atom in the Mott-Hubbard insulating state is determined bythe electronic filling of the atomic levels through specific rules named Hund’s rules.The interactions between those local moments in ordered solids are responsible for the various longrange ordered magnetic states (ferromagnetic or antiferromagnetic) or their absence thereof in the casewhere ordering is prohibited by geometric frustration effects, as will be illustrated later on. So far we have seen that the original properties of electronic matter are mostly governed by the mag-nitude of the Coulomb repulsion between electrons which is essential in the magnetic properties andin promoting localized electronic states rather than extended states. But, although we mentioned itat many places already, we did not consider so far one of the most important correlated electronicstates which has been studied at length during most of the last century, that is superconductivity. Thiselectronic state of matter is by no way an independent electron case, as the basic feature of this state isan electronic organisation which emphasizes pairs of electrons, the Cooper pairs. This has been high-lighted in classical metals by the development of the Bardeen Cooper Schrieffer (BCS) theory whichstates that in the presence of an attractive interaction between electrons, no matter how weak it maybe, the electron gas becomes unstable. 4igure 3: Spatial representation of a Cooper pair. The lattice distortions shown by pink and greenshadings have been produced by the electrons at r2 and r1 respectively and trap the electrons withopposite spins in a singlet state.The beauty of this unexpected physical situation is that the lower energy condensed electronic state isa quantum state of electronic matter in which the correlations between electrons extend on macroscopicdistances.The mystery which prevented the actual understanding of superconductivity during the firsthalf of the 20th century concerned the actual possibility of such an attractive interaction betweenelectrons. This has only been understood when it had been noticed that the electrons do attract theions of the atomic background, and that their displacements (the phonons) being slow due to the largeionic masses provide a memory effect which mediates an attractive interaction between electrons. Ifthat electron-phonon interaction dominates the electronic Coulomb repulsion, then the net attractiveinteraction favors the pairing of electrons which is qualitatively depicted in Fig.3. The pairing ofelectrons results in the many body electronic states which is the basis of the electronic properties ofthe superconductors. One of the main unexpected behaviors which could be explained by the BCStheory is the existence of a gap between the electronic superconducting ground and excited states. Theoccurrence of such a gap has been initially ascertained by NMR experiments.
We have examined so far two completely different limiting descriptions of electronic states in a solid. Inthe band structure approach we have described the case of electrons considered as independent, theirinteractions being restricted to an averaged potential. The delocalisation of these electrons between theatomic sites driven by the transfer integrals may yield metallic states. In contrast we have considered thespecific situation for which electrons localized on ionic states lead to local atomic magnetic moments.Those arise when the Pauli principle and on site inter-electronic Coulomb repulsion are taken intoaccount properly. We have assumed implicitly that these electrons do not delocalise when the transferintegrals between electrons on neighboring ions are small enough in such solids. This then correspondsto an insulating magnetic state quite different from the band insulating states considered so far in theindependent electron band approach.The actual situation in real materials does indeed sometimes correspond to these limiting cases, buta wide variety of solids correspond to intermediate situations, like that of ferromagnetic metals suchas
F e or N i . But the correlated electron physics is now rich with examples of such intermediate cases5hich are quite important both for the fundamental questions raised and for the applications of thenovel physical effects which come into play.
In a Mott-Hubbard insulator, if we increase t (or if we consider compounds with lower values of U ), fora certain critical value of t/U , the upper and lower Hubbard bands begin to overlap (see Fig.2 right),causing the band gap to disappear and leading to a metallic state. Such an increase in t can be producedby bringing the atoms closer together. This was first achieved in the case of doped semiconductors byincreasing the donor concentration, e.g., by increasing the concentration of phosphorus in silicon. Thiscauses the hydrogen-like orbitals of P to move much closer together and increases the hopping integrals,while remaining in a configuration corresponding to one electron per donor atom. A simpler way toachieve this situation directly without changing the number of electrons in a material is to apply anexternal pressure. This increases the hopping integrals t by bringing the atoms closer together, providedthat the material is compressible. In the metallic state thereby induced, one then observes magneticand thermodynamic properties which require taking into account the existence of the strong coulombrepulsion U . As for the Mott-Hubbard insulator, let us point out that it looks at first glance like a bandinsulator, the only difference being that here each Hubbard band contains only Nn states rather than2 N n states in the case of the band theory of section 2. Chemical treatment may be envisaged to change the number of electrons in a Mott insulator. Forexample, it can be doped with holes, reducing the number of electrons in the lower Hubbard band toa number N e smaller than N n . This is exemplified by the case of cuprates such as Y Ba Cu O or La CuO which are antiferromagnetic Mott insulators.In the latter, the Cu are in a 3 d state with spin1 /
2, which order antiferromagnetically below 340K. By chemical exchange of a fraction x of La by Sr one can typically reduce the number of Cu electrons to become N e = (1 − x ) N n . This reductionof the number of electrons in the lower Hubbard band suggests that the doped Mott-Hubbard insulatoris expected to be a metal. Experimental investigations carried out on the cuprates, and also on certainother classes of doped Mott insulators, have shown that doping gradually reduces the N`eel temperatureof the antiferromagnetic state. This AF state is completely suppressed for a low level of doping, of theorder of x ≈ .
05, as can be seen in the phase diagram for La − x Sr x CuO displayed in Fig.4. The importance of the cuprates in the physics of correlated systems has resulted from the discoverythat when the AF is suppressed by hole doping, the doped metallic state which results has a SC groundstate and displays strange metallic and magnetic properties. The most surprising feature has been thefact that the superconductivity discovered in these materials has the highest critical temperatures T c found so far in any superconducting material, and exceeds any T c which could be expected within theBCS approach known to apply in classical metallic states. This has immediately led to the idea that SC in the cuprates has an exotic origin linked with electron-electron interactions rather than the classicalelectron-phonon driven superconductivity which prevails in classical metals. An important observationin the cuprates has been the fact that the phase diagram with increasing hole doping displays a domeshaped SC regime, that is SC disappears for dopings beyond about 0.3.While the cuprates are certainly exotic superconductors, let us state that many other materialshave been shown to display situations where magnetism and SC are proximate to each-other in phasediagrams. In pnictides those are sometimes spanned by doping as in the cuprates, but in other familiesof compounds the phase diagrams are spanned by pressure control of the overlap integrals as for organic,heavy fermions or Cs C compounds. The author shall present many examples of such families in theScholarpedia article NMR in strongly correlated materials[6.6igure 4: Elementary phase diagram of the cuprates obtained by hole doping the AF Mott insulatingstate. The AF state is rapidly destroyed by hole doping beyond 0.05 and opens the way to a metallicstate which becomes superconducting at low T . One observes a SC T c dome shape as a function of thehole doping. The highest T c occurs for optimal doping and one speaks of underdoped and overdopedstates. Such original states have been revealed initially by experimental techniques which were quite adapted atthe time of the discovery of the cuprates to studies of their electronic properties. Among those, NuclearMagnetic Resonance (NMR) is a technique which is quite essential as it permits local measurementsin the materials. This gives precious information which goes beyond the first indications given by themacroscopic magnetic measurements as they permit one to differentiate the properties of the materialswhich can be attributed to specific phases or sites in the structure. Also, as usual for magnetic materials,inelastic and elastic Neutron scattering techniques reveal the occurrence of magnetic responses and oftheir k-dependence.Significant effort has been invested to improve the quality of single crystals which are essential forthe studies of the transport properties in these exotic metals and SC . Static or pulsed high magneticfield sufficiently large to suppress the superconducting state have been achieved, though this not yetpossible for samples with high optimal T c .Other new specific techniques for studies of surfaces of 2D compounds have been developed duringthe last decades. The Angular Resolved Photoemission Spectroscopy (ARPES) uses X-rays generatedby synchrotrons to perform k-space resolved spectroscopy of the occupied electron states. This permitsdetermination of the band structures of these correlated electron materials. Deviations with respect tosimple band calculations permit determination of the incidence and strength of the electronic correla-tions. Also Scanning Tunneling Microscopy experiments reveal spatial inhomogeneities of the gaps andof the electronic structures at surfaces in these materials. Some experimental groups have developedFourier transformations at a level of refinement which allowed them to reproduce some of the ARPESspectral information. The existence of charge density wave transitions is also detected by ResonantInelastic X-ray Scattering (RIXS) or Resonant Elastic X-ray Scattering (REXS).Many of these novel techniques have been improved by recent technical developments, but theirinput on the physics of correlated electron systems are still far from being fully understood at this timeand will be introduced in dedicated Scholarpedia articles[6].7 efereces
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