Inelastic Light Scattering from Correlated Electrons

Thomas P. Devereaux∗

Department of Physics, University of Waterloo, Waterloo, ON, Canada N2L 3G1

Rudi Hackl†

Walther Meissner Institute, Bavarian Academy of Sciences, 85748 Garching, Germany

(Dated: July 16, 2006)

Inelastic light scattering is an intensively used tool in the study of electronic properties of solids.Triggered by the discovery of high temperature superconductivity in the cuprates and by new de-velopments in instrumentation, light scattering both in the visible (Raman effect) and the X-raypart of the electromagnetic spectrum has become a method complementary to optical (infrared)spectroscopy while providing additional and relevant information. The main purpose of the re-view is to position Raman scattering with regard to single-particle methods like angle-resolvedphotoemission spectroscopy (ARPES), and other transport and thermodynamic measurements incorrelated materials. Particular focus is placed on photon polarizations and the role of symmetryto elucidate the dynamics of electrons in different regions of the Brillouin zone. This advan-tage over conventional transport (usually measuring averaged properties) indeed provides newinsights into anisotropic and complex many-body behavior of electrons in various systems. Wereview recent developments in the theory of electronic Raman scattering in correlated systemsand experimental results in paradigmatic materials such as the A15 superconductors, magneticand paramagnetic insulators, compounds with competing orders, as well as the cuprates withhigh superconducting transition temperatures. We present an overview of the manifestations ofcomplexity in the Raman response due to the impact of correlations and developing competingorders. In a variety of materials we discuss which observations may be understood and summarizeimportant open questions that pave the way to a detailed understanding of correlated electronsystems.

Contents

I. INTRODUCTION 2A. Overview 2B. Historical Review 3C. What Can One Learn from Electronic Raman

Scattering? 4D. State of the Art Experimental Technique 6

II. THEORY OF ELECTRONIC RAMANSCATTERING 8A. Electronic Coupling to Light 8

1. General Approach 82. Importance of Light Polarization 11

B. Formalism: Single-Particle Excitations and WeakCorrelations 121. Particle-hole Excitations 122. Im(1/ε) and Sum Rules 143. Intra- vs. Inter-cell Charge Fluctuations 14

C. Formalism: Strong Correlations 151. General Approach to Treating Correlations 152. Correlated Insulators - Heisenberg Limit 16

D. Electronic Charge Relaxation 181. Weakly-Interacting Electrons 182. Impurities 193. Interacting Electrons - Non-Resonant Response 204. Interacting Electrons - Resonant Response 225. Interacting Electrons - Full Response 236. Superconductivity 247. Collective Modes 28

∗Electronic address: tpd@lorax.uwaterloo.ca†Electronic address: hackl@wmi.badw.de

III. FROM WEAKLY TO STRONGLYINTERACTING ELECTRONS 29

A. Elemental Metals and Semiconductors 30

B. Conventional Superconducting Compounds 30

1. A15 compounds 30

2. MgB2 and the Borocarbides 32

C. Charge Density Wave Systems 33

1. 2H−NbSe2 33

2. 1T -TiSe2 34

D. Kondo or Mixed-Valent Insulators 35

E. Magnetic, charge and orbital ordering: Ramanscattering in Eu-based compounds, Ruthenates, andthe Manganites. 35

IV. HIGH TEMPERATURE SUPERCONDUCTINGCUPRATES 37

A. From a Doped Mott Insulator to a Fermi Liquid 39

B. Superconducting Energy Gap and Symmetry 40

1. Symmetry: B1g and B2g 41

2. The A1g problem – Zn, Ni and Pressure 42

3. Resonance Effects 43

4. Electron versus Hole-Doped Materials 44

C. Superconducting Gap: Doping Dependence 45

D. Normal State: Dichotomy of Nodal and Anti-NodalElectrons 48

1. Unconventional Metal-Insulator Transition 48

2. Quantum Critical Point(s) 50

3. Role of Fluctuations and Incipient OrderingPhenomena at small doping 51

V. CONCLUSIONS AND OPEN QUESTIONS 53

Acknowledgements 54

References 55

2

I. INTRODUCTION

A. Overview

Raman scattering is a photon-in photon-out processwith energy transfered to a target material. Most of thelight is elastically scattered from the sample, a fractionis color-shifted and collected at the detector.

Light couples to electronic charge in solids and canscatter inelastically from many types of excitations ina sample, as shown schematically for YBa2Cu3O6.5 inFigure 1. Optical phonons produce sharp peaks at wellknown positions and orientations of the incoming andoutgoing photon polarizations, while a large broad fea-ture centered at much higher energies due to two-magnonscattering occurs in compounds with antiferromagneticcorrelations. This review article largely places empha-sis on the electronic Raman scattering continuum uponwhich the phonons and magnons are superimposed.

Light scatters off of electrons by creating variationsof electronic charge density in the illuminated region ofa sample. By observing the frequency shift and polar-ization change of the outgoing photon compared to theincoming photon, the properties of charge density relax-ation is measured. However, measuring the Raman effectof photons scattering from electrons is a difficult proposalto carry forward. It is made difficult precisely because ofthe coupling of the real photon vector potential and theexchange of virtual photons which mediate the Coulombforces between electrons. In simple metals, variations ofthe charge density will be largely screened by the mo-bile electrons, and the system of electrons responds col-lectively at a characteristic plasma frequency of severalelectron volts. In semiconductors or well-developed bandinsulators, the creation of charge density fluctuations oc-curs only via the population of excited states across aband gap - again on the scale of several electron volts.Therefore it is difficult to investigate the behavior of elec-trons at low energies. In fact, hardly any measurementsof electronic Raman scattering in simple metals exist pre-cisely for this reason, and focus on semiconductors is usu-ally placed on plasma excitations. Since the dynamicsof electrons lying near the Fermi surface govern the be-havior of transport in most systems, this would give theimpression that Raman would have but little to offer insimple metals and insulators.

Yet the Raman effect is extremely well suited to studyelectrons in systems with non-trivial electron dynamics.First well-studied in the context of breaking Cooper pairsin superconductors in the period from 1980 to 1990, thefield of Raman scattering from electronic excitations hasgrown tremendously over the past few decades to studythe evolution of electron correlations in a variety of sys-tems in which many-body interactions are essential tothe physics of new materials and their potential deviceapplications.

Raman spectroscopy has become an indispensable toolin the arsenal for understanding many-body physics. One

of the most celebrated achievements of electronic Ramanscattering has been the ability to focus on the natureof electron dynamics in different regions of the Brillouinzone. This distinguishes Raman scattering from mostother transport and thermodynamic measurements, al-lowing the study of the development of correlations inprojected regions of the Brillouin zone. By simply align-ing the polarization orientations of the incoming andoutgoing photons, charge excitations can be selectivelymapped and analyzed using group-theoretical symmetryarguments. The search for conventional as well as ex-otic excitations in strongly correlated matter has beengreatly enhanced. Raman spectroscopy has provided newand valuable insights into unconventional superconduc-tivity and collective modes, excitations in charge, spinand/or orbitally ordered systems as well as the compe-tition between the various ordered phases. In addition,new insights into electron dynamics of metal-insulatortransitions, quantum phase transitions, and the concomi-tant quantum critical behavior could be obtained. Thepurpose of this article is to review the essence of thesenew developments in a “snapshot” of the current state ofinvestigation.

The overall agenda of the paper is to provide a vehicleto sort through the extensive literature, learn about theoutstanding problems, and become aware of the level ofconsensus. In order to present a detailed picture of thecurrent status of electronic light scattering, other types ofexcitations, such as phonons and magnons, are mainly ig-nored. There have been many reviews on inelastic lightscattering from phonons and magnons. The reader isreferred to earlier reviews by Klein (1982b) for a fun-damental treatment of scattering from phonons, whilestudies of phonons in high temperature superconduc-tors are summarized by Thomsen (1989) and Shermanet al. (2003). Recently Lemmens et al. (2003) and Gozaret al. (2005b) reviewed magnetic light scattering in low-dimensional quantum spin systems and cuprates. Due tospace limitations we cannot give adequate commentaryon these exciting and developing fields.

The outline of our review is as follows. After a brief his-torical summary, the fundamental experimental aspectsand theoretical developments for electronic Raman scat-tering are presented in the first part of the article. A gen-eral treatise on the theory of electronic Raman scatteringis given in Section II with a view toward the formalism forboth weak and strong correlations. Results from model-specific calculations can be found in Section II.D. Read-ers who are more interested in summaries of experimentalwork may want to skim these sections and skip to SectionIII, where a review of Raman scattering measurements ina variety of correlated materials is given with a view to-ward common features manifest from strong correlations.The presentation is generally organized in systems withincreasing complexity of correlations and competing or-ders.

In this framework, the canon of work on the high tem-perature superconductors in the last part of our review is

3

FIG. 1 Characteristic Raman scattering spectrum taken on YBa2Cu3O6.5 showing light scattering from phonons (blue lines),magnons (red), and electrons (green). Courtesy of Matthias Opel.

presented in Section IV. This detailed part of the reviewis organized in conceptual issues of correlations, super-conductivity, normal state properties, and the propensitytoward charge and spin ordering in various families of thecuprates. In all subsections in this part, data on a varietyof cuprate materials are summarized.

The review closes with a general discussion of openquestions for both experimental and theoretical develop-ments in Raman scattering, and points out new directionsin which our understanding of electronic correlations maybe further enhanced.

B. Historical Review

Inelastic scattering of light was discovered indepen-dently in organic liquids by Raman and Krishnan (1928)and in quartz by Landsberg and Mandelstam (1928) whoproperly explained the observed effect: The energy ofthe incoming photon is split between the scattered oneand an elementary excitation in the solid. Shortly there-after in 1930 C. V. Raman was awarded the Nobel prize,and his name was associated with the effect (Fabelinskiı,1998; Ginzburg, 1998; Pleijel, 1930). Although the phe-nomenological description by Smekal (1923) in terms ofa periodically modulated polarizability qualitatively cap-tures the relevant physics including the selection rules,the effect is genuinely quantum mechanical as describedfirst and ahead of the experimental discovery by Kramersand Heisenberg (1925) in context of the dispersion in di-electrics.

Soon after the observation of light scattering from vi-brational excitations, Verkin and Lazarev (1948) andKhaıkin and Bykov (1956) attempted to use the newtechnique for studying electronic excitations. Theypicked one of the most ambitious subjects, i.e., lightscattering from superconducting gap excitations in con-ventional metals. It is not at all surprising that they

could not succeed. In a seminal paper Abrikosov andFal’kovskiı (1961) not only calculated the Raman re-sponse of a typical elemental superconductor but alsodemonstrated that the sensitivity in the early experi-ments was by approximately 5 or 6 orders of magni-tude too low. In 1980 light was finally scattered suc-cessfully from superconducting electrons in 2H − NbSe2

(Sooryakumar and Klein, 1980). Balseiro and Falicov(1980) and Littlewood and Varma (1981, 1982) arguedthat the superconducting excitations in this system be-come Raman active mainly via their coupling to a charge-density wave mode (CDW). After the observation ofgap excitations in the A15 compounds Nb3Sn and V3Si(Dierker et al., 1983; Hackl et al., 1982, 1983; Klein,1982a) it was clear that light can be scattered directlyby Cooper pairs (Dierker et al., 1983; Klein and Dierker,1984). Tutto and Zawadowski (1992) demonstrated thatboth types of coupling contribute.

Collective excitations of normal electrons were first ob-served in semiconductors (Mooradian and Wright, 1966)following theoretical studies by Pines (1963), Platzmanand Tzoar (1964). As a function of doping the plasmonpeak moves across the phonon energies leading to strongelectron-lattice interactions. In heavily doped silicon,with the plasma energy well beyond the vibration spec-trum, the evolution of the phonon line shape (Cerdeiraet al., 1973; Fano, 1961) clearly demonstrated the exis-tence of an electron continuum. In 1977, fluctuationsof electrons between pockets of the Fermi surface of sil-icon were observed by Chandrasekhar et al. (1977) andexplained subsequently by Ipatova et al. (1981). In mag-netic fields, transitions between Landau levels were found(Worlock et al., 1981)1. Strong phonon renormalization

1 The subject has been reviewed in detail by Abstreiter et al.

4

effects also occur in metallic alloys with A15 structure(Schicktanz et al., 1980; Wipf et al., 1978). The originof the broad continuum, which interacts with phononsand is redistributed below the superconducting transi-tion (Klein and Dierker, 1984), is certainly electronic butas of today is still not fully understood.

The full power of the method became apparent afterthe discovery of copper-oxygen compounds (Bednorz andMuller, 1986) with superconducting transition tempera-tures above 100 K. It turned out that, in contrast toinfrared spectroscopy, momentum dependent transportproperties can be measured with Raman spectroscopy,since different regions of the Brillouin zone can be pro-jected out independently by appropriately selecting thepolarizations of the incident and scattered photons (De-vereaux et al., 1994a). New theoretical ideas were notonly applied to the superconducting but also to the nor-mal state. The spectra extend over energy ranges as largeas electron Volts (eV) (Bozovic et al., 1987; Cooper et al.,1988a,b; Kirillov et al., 1988) and are similar to those inthe A15s or in rare earth elements (Klein et al., 1991).Both elastic (Zawadowski and Cardona, 1990) and in-elastic (Itai, 1992; Kostur, 1991; Virosztek and Ruvalds,1992) relaxation of electrons indeed produces light scat-tering over such a broad range of energies. It soon be-came clear that a continuum extending over an eV cannotoriginate from elastic scattering, but only from inelasticprocesses or interband transitions. However, it has notbeen straightforward to pin down the types of interac-tions.

At very low energies, spin- (Yoon et al., 2000) andcharge-ordering fluctuations were reported in manganitesand, respectively, in ladder compounds (Blumberg et al.,2002) and in the cuprates (Venturini et al., 2002b). Theresponse should not be confused with that of an orderedspin/charge-density-wave (SDW/CDW) state (Benfattoet al., 2000; Klein, 1982c; Zeyher and Greco, 2002). Sincea characteristic energy decreases rather than increasesupon cooling (Blumberg et al., 2002; Caprara et al., 2002,2005; Venturini et al., 2002b; Yoon et al., 2000), which isin clear contrast to typical order parameter behavior.

Along with the early studies of charge excitations,Fleury and coworkers observed Raman scattering fromspin waves in antiferromagnetically ordered FeF2, MnF2,and K2NiF4 (Fleury et al., 1966, 1967, 1970). Elliot andLoudon (1963) and Fleury and Loudon (1968) presenteda detailed theoretical description which allowed them tosemi-quantitatively understand the spectral shape andthe cross section. With the discovery of the cuprates byBednorz and Muller (1986) also this field experienced arenaissance in particular for the study at low doping closeto the antiferromagnetic Neel state (Gozar et al., 2004,2005b; Lyons et al., 1988; Sugai et al., 1988; Sulewskiet al., 1991). Raman scattering is probably the most

(1984)

precise method for determining the exchange coupling Jthough the theoretical understanding is still incomplete.In this context, spin-Peierls systems (Loosdrecht et al.,1996) and ladder compounds (Abrashev et al., 1997)shifted very much into the focus of interest (Dagotto,1999).

Very recently, light scattering from “orbitons”, i.e.from a propagating reorientation of orbitals, has beenproposed to explain new modes in the the Raman spec-tra (Saitoh et al., 2001). However, there is no agreementyet on whether or not orbitons can be observed indepen-dent of phonons or other excitations (Choi et al., 2005;Gruninger et al., 2002; Kruger et al., 2004).

C. What Can One Learn from Electronic RamanScattering?

We give now a qualitative introduction into the rela-tionship between Raman spectroscopy and other experi-mental techniques. We begin by drawing the distinctionbetween one-particle and many-particle properties.

Typically, electronic states in solids are characterizedby their energy dispersions as well as the characteristiclifetime of an electron placed into such a state. Thisstate is characterized by the single particle propagatoror Green’s function for the electron,

G(k, ω) =1

ω − ξk − Σ(k, ω). (1)

Here ξk denotes the bare energy band dispersion cal-culated from a solvable model. Σ represents the elec-tron self-energy which encompasses all the informationpertaining to interactions of the single electron in statek to all other excitations of the system. Usually theself energy can only be obtained via approximate meth-ods. Some of these approximations are quite good -such as electron-phonon interactions in metals (knownas Migdal’s approach (Migdal, 1958)) for example, whileothers are more difficult - such as the Coulomb interac-tion between other electrons. The self energy is a com-plex function, Σ = Σ′ + iΣ′′, which, in general, dependson temperature, momentum and energy. The real part ofthe self energy determines how the energy dispersion ξkis renormalized by the interactions while the imaginarypart determines the lifetime of the quasiparticle placedinto the state k.

The spectral function is directly related to the analyti-cally continued electron’s Green’s function for frequencieson the real axis via the replacement iω → ω + iδ

A(k, ω) = − 1π

limδ→0

G′′(k, ω + iδ) (2)

which is measurable via modern angle-resolved photoe-mission (ARPES) techniques and has provided an im-mense amount of information in strongly correlated sys-tems (Campuzano et al., 2002; Damascelli et al., 2003).For non-interacting electrons, A(k, ω) is a δ-function

5

peaked at the pole of the propagator when the fre-quency ω equals the bare band energy ξk. Interac-tions broaden the spectral function and give it non-trivialtemperature and frequency dependences as well as non-trivial anisotropies in momentum-space if the interac-tions among electrons are anisotropic. The spectral func-tion describes real electrons, hence integrals over all en-ergies must obey sum rules, such as (i)

∫dωA(k, ω) = 1

and (ii)∫dωf(ω)A(k, ω) = n(k) with the Fermi Dirac

distribution f(ω) and the momentum distribution func-tion n(k).

If the electronic interactions are weak, one usually usesthe nomenclature of Landau and refers to dressed quasi-particles replacing the electron as the fundamental exci-tation in the solid. These interactions may be character-ized by the residue of the pole (usually denoted by Zk)and the quasiparticle effective mass m∗/mb = (Zk)−1

with mb the bare band mass for quasiparticles lying nearthe Fermi surface. Zk is related to the real part of the selfenergy Σ′ which can be expanded for electrons near theFermi surface as Σ′(k, ω) Σ′

0(k) + ω∂Σ′(k, ω = 0)/∂ω.According to Luttinger’s theorem, the Fermi surface av-erage of Σ′

0(k) vanishes. The enhancement of the qua-siparticle mass over the band mass can be written asm∗/mb = (1 − ∂Σ′/∂ω). One often defines m∗/mb =1+λ with the dimensionless coupling constant λ ≥ 0 (see,e.g., (Ashcroft and Mermin, 1976)). Zk = (1−∂Σ′/∂ω)−1

is always smaller than 1, reflecting the fact that even forω = 0 and T = 0 only a fraction Zk of the spectral weight(coherent part) is in the pole of A(k, ω) while 1−Zk (in-coherent part) is distributed over larger energy scales.Equivalently, Zk ≤ 1 is the discontinuity at kF of thezero-temperature momentum distribution function n(k).If Zk approaches zero (∝ 1/ lnω) the system is referredto as a marginal Fermi liquid (Varma et al., 1989a), andsum rule (i) is exhausted only at energies much largerthan ξk. Thus, knowledge of the self energy is an impor-tant requisite to understanding many-body interactions.

For this reason, single particle methods such asARPES, electron tunneling and specific heat measure-ments have been applied extensively to study correlatedelectron systems. Very much stimulated by the discov-ery of superconductivity in the cuprates (Bednorz andMuller, 1986) ARPES and tunneling spectroscopy havedeveloped more rapidly than any other method in thelast decade. ARPES data has given unprecedented in-sight into momentum-resolved single electron propertiesand their many body effects (Campuzano et al., 2002;Damascelli et al., 2003), while tunneling measurementshave provided information on pairing (Mandrus, 1991;Renner, 1995; Zasadzinski et al., 2002) and have recentlyelucidated many issues of nanoscale inhomogeneities inthe cuprates and their connection to superconductivity(Fang et al., 2006; Hanaguri et al., 2004; Hoffman et al.,2002; Howald et al., 2003; Kivelson et al., 2003; McEl-roy et al., 2003, 2005; Vershinin et al., 2004). Detailedknowledge of phase transitions in the cuprates has beenobtained from specific heat studies (Loram and Tallon,

2001; Moler et al., 1994; Roulin et al., 1998). Due tospace limitations we only have listed some of the later ref-erences of important experimental papers which we hopeserve as an entry point for the reader to search backwardsin time to follow the developments.

Yet knowledge of the spectral function and single-particle excitation spectra do not yield information abouthow the electrons may transport heat, current, entropy,or energy. For this one needs two-particle correlationfunctions for charge or spin which can be measured by,e.g., ordinary and heat transport, optical spectroscopy,neutron and light scattering. As an example for such acorrelation function we consider a standard expressionfor the generalized Kubo susceptibility χ′′

a,b(Ω) of weaklyinteracting, isotropic normal electrons (see, e.g., Mahan(2000)),

χ′′a,b(Ω) =

2V

∑k

akbk

∫ ∞

−∞

dω

πG′′(k, ω)G′′(k, ω + Ω)

× [f(ω) − f(ω + Ω)] . (3)

Here V is the volume, ak, bk are the bare vertices rep-resenting quasiparticle charge (ak = 1) or current (ak =jk = ek) correlation functions, and the factor 2 accountsfor spin degeneracy. The absorptive part of the conduc-tivity σ′ = χ′′

j,j(Ω)/Ω measures essentially a convolutionof occupied and unoccupied states. For electrons weaklyinteracting with impurities the conductivity can readilybe calculated to exhibit a Lorentzian dependence on Ωrepresented by

σ′(Ω) = σ01

1 + (Ωτ)2(4)

where σ0 is the dc (Ω = 0) conductivity, and the relax-ation time τ = −(2Σ′′)−1 controls both the width ofthe spectral function and the conductivity as a functionof frequency.2 A very similar expression is found for lightscattering3. Thus in the case non-interacting electrons,single and two-particle correlation functions give similarresults.

This is also true by and large if weak but essen-tially isotropic interactions lead to an energy dependentΣ′′(ω) and, for causality, to a finite Σ′(ω). Gotze andWolfle (1972) and, more phenomenologically, Allen andMikkelsen (1977) (for a more recent reference see (Basovand Timusk, 2005)) discuss how this generalization mod-ifies the response given by Eq. (4), which is then oftenreferred to as the extended Drude model. However, in-teracting systems require some care. For example, in

2 Note that Eq. (3) does not return the proper transport lifetime τt

which differs by a factor of typically 1− cos θ with the scatteringangle θ since events with θ ≈ 0 do not contribute to the resis-tivity. This deficit must be taken care of by vertex corrections(Mahan, 2000).

3 The case of Raman scattering is described in detail in reference(Opel et al., 2000) and touched upon briefly in section IV.D.1.

6

superconductors, both the single and the two-particleresponses yield the energy gap. Yet two-particle corre-lation functions also have coherence factors which canbe crucially important to determine the gap symmetryin unconventional systems. Generally, collective modes(such as the plasmon or excitons) appear directly in two-particle correlation functions but only indirectly in thespectral function.

Sometimes the results from single- and two-particlemeasurements can be qualitatively different, even fornon-interacting electrons. As an illustration, we considerfirst the metal-insulator transition occurring in a systemof otherwise non-interacting electrons in a disordered en-vironment (Anderson transition). Here, backscatteringof electrons from impurities leads to destructive phaseinterference, and the electrons become localized once acritical concentration of impurities is in place in threedimensions. Thus while the conductivity is critical andvanishes at the metal-insulator transition, the spectralfunction, or equivalently the density of states, is uncrit-ical. This distinction becomes even more pronounced ifthe electron interactions are strong and anisotropic, andthe bare vertices along with the Green’s functions en-tering into Eq. (3) must be renormalized by the stronginteractions.

As a second example, in the spinless Falicov-Kimballmodel light d−electrons strongly interact with localizedf−electrons and are characterized by the Hamiltonian(Falikov and Kimball, 1969)

H = − t∗

2√D

∑〈i,j〉

(c†i cj + c†jci) + Ef

∑i

wi

−µ∑

i

(c†i ci + wi) + U∑

i

c†i ciwi (5)

where c†i (ci) create (destroy) a conduction electron atsite i, wi is a classical variable (representing the local-ized electron number at site i) that equals 0 or 1, t∗ isa renormalized hopping matrix element that is nonzerobetween nearest neighbors on a hypercubic lattice in D-dimensions, and U is the local screened Coulomb inter-action between conduction and localized electrons. 〈i, j〉denotes a sum over sites i and nearest neighbors j. Ef

and µ are adjusted to set the average filling of conductionand localized electrons. This model has been solved ex-actly for electrons on a hypercubic lattice in the limit oflarge coordination number (Freericks and Zlatic, 2003b).The system undergoes a metal-insulator transition (MIT)at half-filling (one electron per site) if the interaction Uis beyond a critical value Uc. On either side of the metal-insulator transition, the density of states is temperatureindependent (van Dongen, 1992), while the conductivityhas a strong temperature dependence (Pruschke et al.,1995) showing the development of the MIT.

In systems with strong and anisotropic interactions,the differences between single and two-particle propertiesare inescapable. This is been borne out in the cupratesby the large amount of work using optical (Homes et

al., 2004) and thermal conductivities (Sutherland et al.,2005), resistivities (Ando et al., 2004), nuclear magneticresonance (NMR) (Alloul et al., 1989) and electron spinresonance (ESR) (Janossy et al., 2003). These experi-ments have revealed basic properties of strongly corre-lated systems and have emerged as key elements to char-acterize the complex behavior of the high-Tc cuprates.

Yet these two-particle measurements are largely insen-sitive of anisotropies, as they measure Brillouin zone av-eraged quantities. As a result they reveal the behavior ofquasiparticles having the highest velocities which, in thecuprates, are the quasiparticles near the nodal regionsof the Brillouin zone. As far as carriers are concerned,the momentum dependence of neutron scattering servesmainly to measure spin dynamics in different regions ofthe Brillouin zone.

In this review, we illustrate that Raman spectroscopygives complementary information to all of these measure-ment techniques, and also may provide detailed informa-tion of charge and spin dynamics of electrons in differentregions of the Brillouin zone. This is due to the polariza-tion selection rules. As with phonon scattering (Hayesand Loudon, 2005), simple group theoretic symmetry ar-guments can be used to focus on electron dynamics indifferent regions of the Brillouin zone. For Raman scat-tering (akbk) is replaced with γ2

k which, in certain limits,may be represented by γk =

∑µ,ν e

iµe

sν∂

2ξk/2∂kµ∂kν ,

with ei,s the incident, scattered light polarization vec-tors.4 Apart from energy-independent scaling factors andvertices with different k dependences there is an extrafactor 1/Ω between Raman and infrared response. It hasbeen shown first by Shastry and Shraiman (1990) and ex-plicitly demonstrated within Dynamical Mean Field The-ory (DMFT) by Freericks and Devereaux (2001) that, un-der certain restrictions, there is a simple correspondencebetween conductivity and Raman response,

Ωσ′(Ω) ∝ χ′′γγ(Ω), (6)

highlighting that electronic Raman scattering measurestransport properties. However, even the simple form ofthe vertices given above shows that a coincidence can beexpected only for an isotropic material. In anisotropicsystems, light scattering can sample parts of the Fermisurface which are unaccessible for infrared spectroscopy.

D. State of the Art Experimental Technique

Over decades Raman scattering was predominantlyused for the study of molecular and lattice vibrationswhich produce isolated and typically narrow lines in thespectra (see Fig. 1). The lines are used as probes whichsensitively react to changes in the environment of the vi-

4 For a microscopic derivation see section II.A.

7

brating atoms. Similar considerations are at the heart ofmagnetic resonance techniques such as NMR and ESR.

If light is scattered from electrons in solids, the spectraare usually continuous (Fig. 1). To study their evolutionas a function of a control parameter, such as temper-ature, doping, magnetic field or pressure, is rather in-volved since the overall shape and not the position ofwell-defined lines matters. In addition, typical cross sec-tions per unit solid angle and energy interval are smallerby several orders of magnitude than those of vibrations.Electronic Raman scattering in a metal typically pro-duces one energy-shifted photon per s, meV, and sr (unitsolid angle steradian) out of 1013 incoming ones. Thelow efficiency is particularly demanding in studies at highpressure since additional losses and complications such asfluorescence and birefringence arise from the windows,which are typically diamond anvils. Although there weresuccessful early experiments (Zhou et al., 1996) the avail-ability of synthetic diamonds brought substantial ad-vances (Goncharov and Struzhkin, 2003).

There are three inventions which finally produced therequired sensitivity: (i) the laser as an intense lightsource providing lines of high spectral purity in a wide en-ergy range, (ii) as an early application of the laser, holo-graphically fabricated gratings without secondary images(ghosts) and an extremely low level of diffusely scatteredlight and, finally, (iii) the invention of charge-coupleddevices (CCD) as a location-sensitive detector with anefficiency at the quantum limit and negligible dark countrate.

Gratings have an extremely well defined number oflines per unit length (cm in the cgs system). This isthe origin of the energy unit cm−1. The following con-versions are frequently used:

1 meV = 11.604 K1 meV = 8.0655 cm−1

kB = 0.69504cm−1

K

Since the CCD has a high spatial resolution down to afew µm it is a superior replacement of the photographicplate. It facilitates recording complete spectra in a singleexposure with energy ranges from meV up to approxi-mately 1 eV, depending on the desired resolution.

The essentials of a setup for inelastic light scatteringwith polarized photons are shown schematically in Fig. 2.The coherent light at energy ωi from the laser (Ar+ andKr+ gas lasers are still very popular) is spatially filtered.A prism monochromator (PMC) selects the desired fre-quency and suppresses incoherent photons from the lasermedium. A combination of a λ/2 retarder and a polar-izer (P1) facilitates the preparation of a photon flux ofa well-defined polarization state and intensity. For ex-citation, the polarization inside the sample counts. Thesame holds for the selection of the proper polarizationfor the scattered photons at ωs. The best results areobtained by using a crystal polarizer (P1, e.g. of Glan-Thompson type) and a Soleil-Babinet compensator for

CCD

LASER

Spectrometer

Sample

PMC

SBP1

P2

/2

/4 /2

Slit

SpatialFilter

SpatialFilter

FIG. 2 Schematic drawing of the light path. The laser lightwith energy ωi is first spatially filtered. A prism mono-chromator (PMC) is to suppress the plasma lines of the lasermedium, only the coherent one at ωi passes the slit. The po-larization is prepared with polarizer P1 and the Soleil-Babinetcompensator (SB). The λ/2 retarder in front of P1 allowsone to adjust the power. Before hitting the sample the lightis once again spatially filtered to maintain an approximatelyGaussian intensity profile in the spot. If the angle of incidenceis not close to zero phase shift effects at the sample surfacemust be taken into account since the polarization inside thesample is important. High speed optics collects the scatteredlight. The polarization state is selected by a λ/4 retarder andpolarizer P2. The λ/2 retarder in front of the entrance slitrotates the light polarization for maximal transmission of thespectrometer (here single stage). Except for the compensatormost retarders and polarizers work also for light propagatingat small angles (up to approximately ±5) with respect to theoptical axis. The configuration shown here is usually referredto as back-scattering geometry since incoming and outgoingphotons have essentially opposite momenta in the sample.

the incoming light and an achromatic λ/4 retarder andanother crystal polarizer (P2) for the scattered light. Inthis way all states, including circularly polarized ones,can be prepared. The λ/2 retarder in front of the en-trance slit of the spectrometer rotates the polarizationinto the direction of highest sensitivity.

Since we wish to discriminate between the 1010−15 elas-tically scattered photons at ωi and the few Raman pho-tons at ωs at very small shifts Ω = |ωi − ωs| <1 meV, a single monochromator is insufficient. A moderninstrument for Raman scattering in metallic samples hasthree stages consisting of essentially independent gratingmonochromators. The first two are usually subtractivelycoupled and select a band from the spectrum of inelas-tically scattered photons. The third stage disperses theband transmitted through the two stages of the premono-chromator into a spectrum which is recorded by the CCD.In this configuration, the dispersion is given only by thethird stage while the first two discriminate the elasticallyscattered laser light. If all stages are coupled additively,the resolution is improved by a factor of 3. Because oflosses at the mirrors and gratings, only 15 to 20 % of thephotons entering the spectrometer arrive at the detector.

8

Since very interesting physics is going on at energieseven below 1 meV (see, e.g., section IV.D.3) the discrim-ination is a cardinal point. Out of the two options onlythe premonochromator gives satisfactory results below10 meV. The price one has to pay is a loss of intensityof approximately 60 %. Alternatively, for energy shiftsabove 10 meV an interferometric notch filter can be used.The latter device is widely used for commercial applica-tions which develop rapidly since the introduction of theCCD. The main fields are quality control and analytics.

At finite temperatures, T > 0, inelastically scatteredlight is found on either side of ωi. As a consequence oftime-reversal symmetry and for phase space argumentsthe energy gain (Anti-Stokes) and loss (Stokes) spectraare related by the principle of detailed balance (equiva-lent to the fluctuation-dissipation theorem) (Landau andLifshitz, 1960; Placzek, 1934)

NAS

NST

=(ωi + Ωωi − Ω

)2

exp(− ΩkBT

)(7)

with NST(AS) and Ω the rate of photons per unit timecollected on the Stokes (Anti-Stokes) side and the energytransferred to the system, respectively. Equation (7) canbe used to determine the temperature of the laser spot.

If spectra are measured in large energy ranges, the sen-sitivity of the instrument has to be taken into account.To this end, the spectral response of the whole system,including all optical elements between the sample and theentrance slit of the spectrometer, the spectrometer itself,and the detector, must be calibrated. This is best doneby replacing the sample with a continuous light sourceof the same size as the laser spot with a known spectralemissivity. A continuous source is of crucial importancefor including the energy dependence of the dispersion inaddition to the bare transmission. In addition, the fre-quency dependence of the sample’s index of refraction,√ε = n+ ik, requires attention in order to get the inter-

nal cross section.The main limitations of present commercial systems

come from geometrical aberrations of the spectrometeroptics and from the relatively low total reflectivity ofthe large number of mirrors. It is a matter of resourcesto improve these caveats. Recently, an improved typeof triple spectrometer with aspherical optics has beendescribed by Schulz et al. (2005). CCDs and gratings areclose to the theoretical limits.

For many studies, light sources with continuously ad-justable lines in an extended energy range would be de-sirable. This holds particularly true for organic materi-als (e.g. carbon nanotubes or proteins) which have rel-atively sharp resonances in the visible and the ultravi-olet. The synchrotron, free electron “lasers”, as well asdye and solid-state lasers are developing rapidly and willgain influence on the field of Raman spectroscopy in thenear future. The same holds for near-field techniques(Hartschuh et al., 2003) which are capable of improvingspatial resolution by at least an order of magnitude belowthe diffraction limit.

II. THEORY OF ELECTRONIC RAMAN SCATTERING

A. Electronic Coupling to Light

The aim of this section is to formulate the theoreticaltreatments for inelastic light scattering in general. Muchhas been done in the development of theories for Ramanscattering, particularly in semiconductors and supercon-ductors. Excellent and early reviews of electronic Ramanscattering have been given by Klein (1983) and Abstreiteret al. (1984) focusing on semiconductors. More recentreviews by Devereaux and Kampf (1997) and Shermanet al. (2003) have focused on theory in superconductorswith applications towards the cuprates. We outline thegeneral formalism for treating systems with weak andstrong correlations and return to a discussion of varioustheoretical models in connection with materials in thefollowing section.

1. General Approach

We first consider a Hamiltonian for N electrons cou-pled to the electromagnetic fields (Blum, 1970; Pines andNozieres, 1966):

H =N∑i

[pi + (e/c)A(ri)]2

2m+HCoulomb +Hfields, (8)

where p = −i∇ is the momentum operator, e isthe magnitude of the elementary charge (the electroniccharge is qe = −e) and c the speed of light. A(ri) is thevector potential of the field at space-time point ri andm the electron mass. HCoulomb represents the Coulombinteraction and Hfields the free electromagnetic part. Weuse the symbol A to denote operators. We expand thekinetic energy to obtain

H = H ′ +e

2mc

∑i

[pi · A(ri) + A(ri) · pi

]

+e2

2mc2∑

i

A(ri) · A(ri), (9)

with H ′ = H0 + Hfields and H0 = (1/2m)∑

i p2i +

HCoulomb. Generally we choose |α〉 to denote eigenstatesof H0 with eigenvalues Eα: H0|α〉 = Eα|α〉. The eigen-state is labeled by all the relevant quantum numbers forthe state, such as combinations of band index, wavevec-tor, orbital and/or spin quantum numbers, for example.The eigenstates may be considered to be Bloch electronswhen the electron-ion interaction is included in H0, asplane-wave states if it is neglected, or may represent Hub-bard states if HCoulomb is taken to include short-rangeHubbard-like interactions between electrons.

The electromagnetic vector potential can be expandedinto Fourier modes A(ri) =

∑q e

iq·riAq. In secondquantized notation, the electromagnetic field operator

9

takes the form (Mahan, 2000)

Aq =

√hc2

ωqV[eqa−q + e∗qa

†q], (10)

with V the volume and a†q, aq are the creation, anni-hilation operators of transversal photons with energyωq = c|q| having a polarization direction denoted bythe complex unit vector eq.

Electronic Raman scattering measures the total crosssection for scattering from all the electrons illuminatedby the incident light. The differential cross section isdetermined by the probability that an incident photonωi is scattered into a solid-angle interval between Ω andΩ+dΩ and a frequency window between ωs and ωs+dωs.A general expression for the differential light scatteringcross-section is given via the transition rate R of scatter-ing an incident (qi, ωi, e

(i)q ) photon into a outgoing state

(qs, ωs, e(s)q ),

∂2σ

∂Ω∂ωs= r20

ωs

ωiR. (11)

Here, r0 = e2/mc2 is the Thompson radius, and R isdetermined via Fermi’s Golden Rule,

R =1Z∑I,F

e−βEI |MF,I |2 δ(EF − EI − Ω), (12)

with β = 1/kBT , Z the partition function, and MF,I =〈F |M |I〉 where M is the effective light scattering opera-tor. The sum represents a thermodynamic average overpossible initial and over final states with k vectors in thesolid angle element dΩ of the many-electron system hav-ing energies EI , EF , respectively. Here Ω = ωi − ωs isthe transferred frequency and we denote q = qi −qs thenet momentum transfered by the photon. MultiplyingEq. (11) by the incident photon flux gives the number ofscattered photons per second into the solid angle incre-ment dΩ within the frequency range dωs, while multiply-ing Eq. (11) by ωs/ωi gives the power scattering crosssection (Klein, 1983).5

From here on we consider the case relevant to Ramanscattering in the visible range with photon energies oftypically 2 eV. Since the momentum transferred to theelectrons, q ∼ 1/δ, with δ the skin depth at the lightenergies (Abrikosov and Fal’kovskiı, 1961), is much less

5 We note that Eqs. (11) and (12) describe scattering inside thematerial. Trivial (Fresnel-formulas) and non-trivial (qz integra-tion) (Abrikosov and Fal’kovskiı, 1961; Fal’kovskiı, 1990, 1991)transformations, which we do not discuss here, are required tofully describe the cross section outside. The qz integration origi-nates from the lack of momentum conservation perpendicular tothe surface of an absorbing medium and can change the spectraqualitatively. From here on, Ω is always the energy transferredto the system.

than the relevant momentum scale of order kF, the Fermimomentum in metallic systems, the limit q → 0 is a goodapproximation in practically all cases.6 However, finiteq should be considered if incident light from frequency-doubled or synchrotron radiation is used where transi-tions between initial and final states at finite q can beprobed. Then the structure of the Landau particle-holecontinuum in weakly correlated systems or transitionsacross a finite q Mott gap in strongly correlated insula-tors can be studied.7

MF,I has contributions from either of the last threeterms in Eq. (9): the first two terms coupling the elec-tron’s current to a single photon and the third term cou-pling the electron’s charge to two photons. This is shownin the schematic cartoon in Figures 3 and 4. Here we con-sider two bands - one partially filled and the other com-pletely filled - in which the incident photon excites anelectron from either the partially or the completely filledband, shown in Figures 3 and 4, respectively. In the non-resonant intraband case, the photon gives up part of itsenergy to leave behind a particle-hole pair, while in theinterband case, an intermediate state is involved, whichdecays via a particle from the partially filled band intothe hole left behind in the filled band. The latter scat-tering may be resonant if the incident or emitted photonenergy corresponds to that of the energy gap separation,otherwise it is non-resonant. In this simple cartoon, onecan see that excitations lying near the Fermi surface arepredominantly probed by non-resonant intraband scat-tering while excitations involving transition between dif-ferent bands - such as the lower and upper Hubbard bandfor example - are probed by intermediate state scattering.The Feynman diagrams representing these contributionsto MF,I are shown in Figure 5.

Referring to Eq. (9), the current coupling has odd spa-

INITIAL STATE FINAL STATE

FIG. 3 Cartoon showing light scattering via non-resonant in-traband scattering.

6 The applicability of the q = 0 limit is discussed in more detail atthe beginning of section III.

7 For a review of relevant work in this regard, the reader is referredto references (Devereaux et al., 2003a,c; Kotani and Shin, 2001;Platzman and Isaacs, 1998).

10

tial symmetry and involves single photon emission or ab-sorption, while the second term is even in parity andinvolves two photon scattering of emission followed byabsorption and vice-versa. The cross-section or the tran-sition rate is thus determined via Fermi’s Golden rule bythe square of the matrix elements shown in Figure 5.

The resulting Feynman diagrams of the contributionsto the cross section are shown in Figure 6. However, not

INTER. STATE FINAL STATEINITIAL STATE

FIG. 4 Cartoon showing light scattering via interband tran-sitions.

FIG. 5 Feynman diagrams contributing to the effective lightscattering operator M . The top diagram represents single-photon absorption, the second two photon scattering, whilephoton emission (absorption) followed by absorption (emis-sion) is shown in the 3rd (4th) diagram from top. The panelson the right are the time-reversed partners of the left dia-grams.

all of them give rise to inelastic light scattering. Someof these terms vanish either because they represent con-tributions to the renormalized photon propagator (Fig-ure 6 (a)), or due to parity arguments (Figures 6 (b)-(d))in the limit of small q scattering. The remaining termscan be classified as non-resonant (Figure 6 (e)), resonant(Figure 6 (h)-(j) and mixed terms (Figure 6 (f)-(g)), sincein the former case the initial and final states must share alarge sub-set of quantum numbers, while the other termscan involve transitions through intermediate states wellseparated in energy and distinct from the initial and fi-nal states. However, we remark that the response is onlytruly resonant if the photon energies are tuned to the en-ergy gap between intermediate and initial or final states.

To obtain a general expression for the matrix elementMF,I for Raman scattering, we use second quantizednotation for the fermions in which the single-particlewave function and it’s conjugate are given by ψ(r) =∑

α cαϕα(r) and ψ†(r) =∑

α c†αϕ

∗α(r), with ϕ,ϕ∗ the

eigenstates of the Hamiltonian H0. Electron states α, βare created, annihilated by c†α, cβ respectively, and theindices refer to the quantum number associated with thestate, such as the momenta and/or spin states. The ma-

FIG. 6 Feynman diagrams contained in the cross-section. (a)gives a renormalized photon propagator in the solid, while (b)-(d) vanish due to parity and thus (a)-(d) do not contributeto inelastic light scattering. Of the remaining contributions,(e) refers to intra-band (sometimes “non-resonant”) scatter-ing within a single band, (f) and (g) are referred to as “mixed”contributions while (h)-(j) describe transitions within a singleor between different bands via intermediate band states. Ifthe light energy is equal or close to the energy difference ofthe states involved resonance effects with a strong enhance-ment of the cross section occur. Therefore the contributionsthemselves are sometimes referred to as “resonant”.

11

trix element MF,I can thus be written as

MF,I = ei · es

∑α,β

ρα,β(q)〈F ∣∣c†αcβ∣∣ I〉+

1m

∑ν

∑α,α′,β,β′

pα,α′(qs)pβ,β′(qi)

×( 〈F ∣∣c†αcα′

∣∣ ν〉〈ν ∣∣∣c†βcβ′

∣∣∣ I〉EI − Eν + ωi

+〈F∣∣∣c†βcβ′

∣∣∣ ν〉〈ν ∣∣c†αcα′∣∣ I〉

EI − Eν − ωs

). (13)

Here |I〉, | F 〉, |ν〉 represent the initial, final and inter-mediate many-electron states having energies EI,F,ν , re-spectively. The many-electron states could be labeled byband index and momentum as, for example, for Blochelectrons. They may also consist of core and valenceelectrons on selected atoms for x-ray scattering, or mayrepresent states of the many-band Hubbard model forcorrelated electrons. ρα,β(q) =

∫d3rϕ∗

α(r)eiq·rϕβ(r) =〈α ∣∣eiq·r∣∣β〉 is the matrix element for single-particle den-sity fluctuations involving states α, β. The momen-tum density matrix element is given by pα,β(qi,s) =〈α ∣∣p · ei,se

±iqi,s·r∣∣ β〉. The first term in the expressionarises from the two-photon scattering term in Eq. (9)in first order perturbation theory. The remaining termsarise from the single-photon scattering term in Eq. (9)in second order via intermediate states ν and involve dif-ferent time orderings of photon absorption and emission.The p · A coupling does not enter to first order since theaverage of the momentum operator is zero.

2. Importance of Light Polarization

At this point, little progress can be made in evaluatingthe matrix elements for Raman scattering without speci-fying the quantum numbers of the electronic many-bodystates. Yet, from Eq. (13), one can apply symmetry ar-guments to view what types of excitations can be createdby the incident photons. In this subsection, we employa general set of symmetry classifications and put specificemphasis on models in later subsections.

The first term in Eq. (13) only arises if the incident andscattering polarization light vectors are not orthogonal,as electronic charge density fluctuations are created anddestroyed along the polarization directions of the incidentand scattered photons. Thus, for instance, this term doesnot probe electron dynamics in which the charge densityrelaxes in a direction orthogonal to the incident polariza-tion direction.

In the limit of small momentum transfer q → 0, thematrix element simplifies to ρα,β(q → 0) = δα,β and thusthis term gives rise to scattering from fluctuations of theisotropic electronic number density.

The remaining terms in Eq. (13) have contributions re-gardless of photon polarization directions. However, one

can further classify scattering contributions by separat-ing the sum over intermediate states |ν〉 into states whichshare some quantum numbers with the initial many-bodystate, such as band index, and states which do not. Sincethe photon momenta are much smaller than the relevantelectron momenta, contributions of the terms where theintermediate states include the initial states are roughlya factor of vF/c smaller than the first term in Eq. (13)and can be neglected (Pines and Nozieres, 1966; Wolff,1966). However contributions where |ν〉 includes higherbands cannot in general be neglected, particularly if theenergy of the incident or scattering light lies near the en-ergy of a transition from the initial state EI to an inter-mediate state Eν . These terms thus give rise to “mixed”and “resonant” Raman scattering.

The polarization dependence of Raman scattering canbe generally classified using arguments of group theory.In essence, the charge density fluctuations brought aboutby light scattering are modulated in directions deter-mined by the polarizations of the incident and scatteredphotons. These density fluctuations thus have the sym-metry imposed on them by the way in which the lightis oriented, and the charge density fluctuations obey thesymmetry rules governing the scattering geometry. Thisis manifest in the dependence of the Raman matrix ele-ments on the initial and final fermion states. In general,the Raman matrix element MF,I = Mα,β

I,F eαi e

βs can be de-

composed into basis functions of the irreducible pointgroup of the crystal Φµ (Devereaux, 1992; Hayes andLoudon, 2005; Klein and Dierker, 1984; Monien and Za-wadowski, 1990; Shastry and Shraiman, 1991)

MF,I(q → 0) =∑

µ

MµΦµ, (14)

with µ representing an irreducible representation of thepoint group of the crystal. Which set of µ contributesto the sum is determined by the orientation of incidentand scattered polarization directions. As an example, ifwe consider the D4h group of the tetragonal lattice, as inthe cuprates, the decomposition can be written as

MF,I =12O

A(1)1g

(exi e

xs + ey

i eys)

+12O

A(2)1g

(ezi e

zs)

+12OB1g (ex

i exs − ey

i eys)

+12OB2g (ex

i eys + ey

i exs )

+12OA2g (ex

i eys − ey

i exs )

+12O

E(1)g

(exi e

zs + ez

i exs )

+12O

E(2)g

(eyi e

zs + ez

i eys), (15)

with Oµ the corresponding projected operators and eαi,s

the light polarizations. This classification demonstrates

12

X M

2gBX M

1gB

FIG. 7 Schematic weighting of the light scattering transitionfor polarization orientations transforming as B1g and B2g fora D4h crystal. High symmetry points are indicated. Here atypical Fermi surface for optimally doped cuprates is repre-sented by the solid line, and the orientations of the incidentand scattered polarization light vectors are shown with re-spect the to copper-oxygen bond directions.

that there is no mixing of representations for q = 0, i.e.the correlation functions read R ∼ 〈O†

µOµ′〉 = Rµδµ,µ′

and there are 6 independent correlation functions, eachselected by combinations of polarization orientations.

Following Shastry and Shraiman (1990), we list inTable I some common experimentally used polarizationgeometries in relation to the elements of the transitionrate R selected. One observes that a complete charac-terization of M can be made from a subset of the polar-ization orientations listed in the table. However, addi-tional polarization orientations can be useful to calibratedata and compare symmetry decompositions from differ-ent combinations of orientations. For geometries withpolarizations in the (a− b) plane in D4h crystals, the ir-reducible representations cannot be accessed individuallyand must be separated by proper subtraction procedures.As a minimum set, four independent configurations arerequired, while additional polarizations may be used forconsistency checks.

In addition, we have listed in Table I the representativebasis functions Φµ(k) taken from the complete set of Bril-louin zone (BZ) harmonics for D4h space group (Allen,1976). This directly points out the connection betweenpolarizations and the coupling of light to electrons. Byvirtue of the k-dependence of the light scattering transi-tion rate, excitations on certain regions of the BZ can becorrespondingly projected out by orienting the incidentand scattered light polarization vectors. Thus, Ramanis one of the few spectroscopic multi-particle probes (theother being inelastic X-ray scattering) able to examinecharge excitations in different regions of the BZ.

For example, as demonstrated in Figure 7, for crossedpolarizations transforming as B1g, light couples to chargeexcitations along the BZ axes (kx or ky = 0), while forB2g, excitations along the BZ diagonals (kx = ±ky)are projected accordingly. Operators like OA2g can-not be accessed independently by linear polarizationsalone. Only sums including circular polarizations al-low the isolation of A2g components. Light scatteringin this orientation can be coupled to chiral excitations.These important symmetry classifications have been ex-

tremely useful to point out anisotropic electron dynamicsin correlated insulators (Devereaux et al., 2003a,c; Shas-try and Shraiman, 1990), superconductors (Devereauxet al., 1994a), disordered (Devereaux, 1992; Zawadowskiand Cardona, 1990) and correlated metals (Einzel andManske, 2004; Freericks and Devereaux, 2001; Freerickset al., 2001). More recently they have been related tosum rules referring to BZ-projected potential energies incorrelated systems (Freericks et al., 2005). In the remain-ing part of this review, such symmetry classifications willbe featured prominently.

B. Formalism: Single-Particle Excitations and WeakCorrelations

We now consider specific cases where simplificationscan be made to Eq. (13). First we assume that the inter-mediate many-particle states only differ from the initialand final states by single electron excitations. This is ex-act in the limit of non-interacting electrons, yet ignoresthe effects of many-body correlations, and specificallythe role of Coulomb interactions on the reorganizationof the initial state into the intermediate states by creat-ing many-particle excitations. We now focus on weaklyinteracting systems, where single-particle excitations arerelatively well defined, and discuss strongly correlatedsystems in later sections.

1. Particle-hole Excitations

Eq. (13) can be simplified by replacing Eν in the de-nominators by EI −Eβ′ +Eβ and EI −Eα′ +Eα in thefirst and second terms, respectively, and use the closurerelation

∑ν |ν〉〈ν| = 1. Commutator algebra eliminates

the four fermion matrix element,

MF,I =∑α,β

γα,β〈F | c†αcβ | I〉, (16)

where

γα,β = ρα,β(q)ei · es +1m

∑β′

(ps

αβ′piβ′β

Eβ − Eβ′ + ωi

+pi

α,β′psβ′,β

Eβ − Eβ′ − ωs

). (17)

Specifying to states α, β indexed by momentum quantumnumbers (such as Bloch electrons), from Eq. (11), theRaman response simplifies to a correlation function S ofan effective charge density ρ,

∂2σ

∂Ω∂ωs= r20

ωs

ωiS(q, iΩ → Ω + i0). (18)

13

Geometry ei es R Basis Functions Φµ(k)

xx, yy x, y x, y RA1g + RB1g12[cos(kxa) + cos(kya)] ± 1

2[cos(kxa) − cos(kya)]

x′x′ 1√2(x + y) 1√

2(x + y) RA1g + RB2g

12[cos(kxa) + cos(kya)] + sin(kxa) sin(kya)

x′y′ 1√2(x + y) 1√

2(x − y) RB1g + RA2g

12[cos(kxa) − cos(kya)][1 + sin(kxa) sin(kya)]

xy x y RB2g + RA2g sin(kxa) sin(kya)1 + 1

2[cos(kxa) − cos(kya)]

LR 1√2(x + iy) 1√

2(x + iy) RB1g + RB2g

12[cos(kxa) + cos(kya)] + sin(kxa) sin(kya)

LL 1√2(x + iy) 1√

2(x − iy) RA1g + RA2g

12cos(kxa) + cos(kya) + [cos(kxa) − cos(kya)] sin(kxa) sin(kya)

xz x z RE1g sin(kxa) sin(kzc)

yz y z RE1g sin(kya) sin(kzc)

zz z z RA

(2)1g

cos(kzc)

TABLE I Elements of the transition rate R for experimentally useful configurations of polarization orientations (given in Portonotation) along with the symmetry projections for the D4h point group relevant for the cuprates. Here we use notations inwhich x and y point in directions along the Cu-O bonds in tetragonal cuprates, while x′ and y′ are directions rotated by 45. Land R denote left and right circularly polarized light, respectively. In our convention left circular light has positive helicity. (Ina right-handed system the polarization rotates from x to y while the wavefront travels into positive z direction by λ/4.) Notethat in back-scattering configuration (see Fig. 2) with ei,s pinned to the coordinate system of the crystal axes the representationfor incoming and outgoing photons with circular polarizations change sign in order to maintain the proper helicity.

Here the Raman effective density-density correlationfunction is

S(q, iΩ) =∑

I

e−βEI

Z∫dτeiΩτ 〈I | Tτ ρ(q, τ)ρ(−q, 0) | I〉,

(19)Tτ is the complex time τ ordering operator and

ρ(q) =∑k,σ

γ(k,q)c†k+q,σck,σ. (20)

The scattering amplitude γ is determined from the Ra-man matrix elements and incident/scattered light polar-ization vectors as

γ(k,q) =∑α,β

γα,β(k,q)eαi e

βs , (21)

with

γα,β(k,q) = δα,β +1m

∑kν

[ 〈k + q | pβs | kν〉〈kν | pα

i | k〉Ek − Ekν + ωi

+〈k + q | pα

i | kν〉〈kν | pβs | k〉

Ek+q − Ekν − ωs

]. (22)

Here, pαi,s = pαe±iqi,s·r. The effective dynamical density-

density correlation function or Raman response S can bewritten in terms of a dynamical effective density suscep-tibility χ via the fluctuation-dissipation theorem,

S(q,Ω) = −π−11 + n(Ω, T )χ′′(q,Ω),

with n(Ω, T ) the Bose-Einstein distribution and

χ(q,Ω) = 〈[ρ(q), ρ(−q)]〉Ω. (23)

Thus, for non-interacting electrons, the Raman responseis given as a two-particle effective density correlation

function and can be calculated easily using, e.g., diagram-matic techniques or via the kinetic equation (Devereauxand Einzel, 1995a). This reduces to evaluating the bubblediagram depicted in Figure 6 with vertices γ dependingupon the incident and scattered photon frequencies.8

The vertex γ depends on polarization, but does notdepend sensitively on q for q << kF. This can be mademore obvious if we consider the sum over intermediatestates kν in Eq. (22). The sum over intermediate statesincludes both the band index of the states created fromthe initial state (i.e. the conduction band) as well asintermediate states separated from the conduction band.The matrix elements of the former are proportional to themomentum transferred by the photon, which in the limitq << kF are smaller by a factor of (vF/c)2 than the otherterms, with vF (c) the Fermi (light) velocity, and can beneglected (Pines and Nozieres, 1966). For the remainingsum over the intermediate states separated from the con-duction band, we assume that ωi,s <<|Ekν − Ek | andrecover the widely used effective mass approximation9

γα,β(k, q → 0) =12

∂2Ek

∂kαkβ. (24)

The symmetry classifications listed in Table I thus canconnect excitations created by certain polarization orien-

8 We remark that technically these expression must be modified ifa resonant condition is satisfied. In that case one needs to ex-pand to higher terms in the vector potential to capture resonanceeffects as perturbation theory breaks down. Yet for low energyRaman scattering, in most cases this is not crucially importantsince in real materials the intermediate states reached via a directresonance are quite broadened by interactions and the resonantterms are not orders of magnitude larger than the non-resonantterms. Thus this treatment may be of more general utility.

9 This is derived in Appendix E of Ashcroft and Mermin (1976).

14

tations to properties of the band structure. Yet it mustbe kept in mind that this connection can only be madein the limit of small ωi,s.

2. Im(1/ε) and Sum Rules

We consider first the case when the incident polariza-tion is parallel to the scattered polarization and the bandEk is isotropic and parabolic, as for the electron gas orlightly-filled isotropic band metal. The scattering ampli-tude γ is then independent of k and the effective densityρ(q) is simply proportional to the pure charge densityρ. Using the definition of the complex dielectric function(Mahan, 2000; Pines and Nozieres, 1966)

ε(q,Ω) = 1 + vqχsc(q,Ω) (25)

is obtained with vq the bare Coulomb interaction, andχsc is the screened or irreducible polarizability. It is de-termined from the full polarizability χ via

χsc(q,Ω) =χ(q,Ω)

1 − vqχ(q,Ω), (26)

but is most easily identified diagrammatically as allcontributions to the polarizability which are irreduciblewith respect to the interaction. The dynamical density-density correlation function (or structure factor) followsas

S(q,Ω) =1πvq

[1 + n(Ω)] Im[

1ε(q,Ω)

]. (27)

The Raman response S is proportional to S with a con-stant of proportionality determined by the ratio of effec-tive to free electron mass.

Inelastic light scattering occurs via the creation ofcharge fluctuations inside the unit cell which are cou-pled via the Coulomb interaction to charge fluctuations inother unit cells. These intercell excitations are thereforewell screened by the Coulomb interaction and reduce thescattering cross section at small q. In particular, for smallq, χ(q,Ω) ∼ NFv

2Fq

2/Ω2, and the response is governed bythe plasma frequency. The Raman response thus obeysthe longitudinal sum rule resulting from particle-numberconservation (Pines and Nozieres, 1966),∫ ∞

0

dΩ Ω Im

[1

ε(q,Ω)

]=π

2Ω2

pl =2π2Ne2

m, (28)

where Ωpl is the plasma frequency and N is the numberof electrons of massm and charge −e in the system. Thusthe only contribution for q → 0 comes from exciting theplasmon being the only charge excitation available forlight scattering at small q in a free-electron gas.

3. Intra- vs. Inter-cell Charge Fluctuations

In the more general case of light scattering in solids,the Raman response may have other contributions com-ing from intra-cell charge fluctuations, provided the band

structure is non-parabolic, as pointed out by Platz-man (1965) and Wolff (1968). Then light can createanisotropic charge fluctuations which are zero on aver-age inside the unit cell and thus are not screened viathe long-range Coulomb interaction, as pointed out byAbrikosov and Genkin (1973). The general expressionfor the screened Raman response function χsc

γ,γ can bewritten as (Devereaux and Einzel, 1995a; Monien andZawadowski, 1990)

χscγ,γ = χγ,γ − χγ,1χ1,γ

χ1,1+χγ,1χ1,γ

χ21,1

χsc, (29)

where χsc = χ1,1(1 − vqχ1,1)−1. This is an exact expres-sion, where the subscript γ denotes the effective Ramandensity and 1 denotes the pure charge density, obtainedwhen the usually momentum dependent vertex γ is re-placed by a constant. The respective χ-s describe thefull density-density, density-Raman density, and Ramandensity-Raman density susceptibilities which are againeach irreducible with respect to the interaction.

The first term in Eq. (29) is the bare response for aneutral system, and the other terms represent the back-flow needed to enforce particle number conservation ofcharge density fluctuations and gauge invariance. Theseterms are important for light scattering configurationswhich transform according to the symmetry of the lat-tice, such as A1g in D4h crystals. In particular, if weconsider scattering from pure charge density fluctuationswhere γ is a constant independent of momentum, whichis an A1g representation, the first two terms in Eq. (29)cancel and χsc and Eq. (27) are recovered. This is in-escapable for q = 0, since then the scattering operatoris given in terms of the total density of electrons, whichcommutes with the bare Hamiltonian and therefore can-not give inelastic scattering channels to light. On theother hand, if we consider the scattering vertex γ to de-pend on wavevector, the backflow terms are not capableof completely canceling the bare Raman response.

Momentum dependence of the vertex γ is quite gen-eral for electrons in solids. In particular, for crossedlight polarizations projecting out representations of lowersymmetry than that of the lattice χγ,1 is identically zeroby symmetry for q = 0 and the backflow terms makeno correction to the Raman cross-section. This occursfor B1g, B2g, and Eg scattering geometries in D4h sys-tems such as the cuprates, for example. While there isno conservation law for light scattering from the exci-tations created by crossed polarizations (Kosztin, 1991),there are sum rules which relate the Raman intensity tomodel-dependent potential energies projected in differentregions of the BZ (Freericks et al., 2005).

15

C. Formalism: Strong Correlations

1. General Approach to Treating Correlations

Section II.B outlined a general approach to inelasticlight scattering when the intermediate states differ fromthe initial and final states only by individual single elec-tron energies. This holds in the limit of weakly interact-ing electrons. In this section we show that the formal-ism is also valid in the Heisenberg limit of the Hubbardmodel including the manifold of zero and 1 doubly oc-cupied sites. These are two limits in which either thekinetic energy or the potential energy of the electrons isdominant. However, the more general case of interest tomost systems is tackling the problem when both kineticand potential energies are roughly equal. The interac-tions are sufficient to give broad spectral functions asmeasured by ARPES, where the incoherent part of thespectral function is manifest from the strong many-bodyinteractions mixing individual electron states.

In this situation, one must resort back to Eq. (13), andcorrelation functions involving two, three and four par-ticles are needed, as depicted in Figure 6. Yet usuallyone is not interested in treating many-body correlationsover various bands and puts focus on a few bands close tothe Fermi level having strong correlations. For examplein the cuprates, one usually takes downfolded Hamilto-nians involving only Cu 3dx2−y2 and O 2px,y orbitals,with short-range Coulomb interactions for two electronsin on-site or neighboring orbital states. The downfoldingprocedure, such as that described by Lowdin (1951), re-moves all other bands (effectively moving them infinitelyfar away in energy from the focus bands), such as theapical oxygen, Cu 4s and other Cu d orbitals, and treatsthe electrons in the bands of interest as having renormal-ized energy dispersion. In the cuprates the downfoldingresults in either a few bands from local density approxi-mation (LDA) approaches in one case or cell perturbationtheory in the strongly interacting case. Thus we considera downfolded tight-binding Hamiltonian

H0 =∑

〈i,j〉,σti,jc

†i,σcj,σ +Hint, (30)

where ti,j are the effective hopping integrals resultingfrom the downfolding procedure, and Hint describes therelevant interactions. For example, we may consider asquare lattice of electrons with strong on-site repulsion Umuch greater than the electron hopping t in the Hubbardmodel,

H = −t∑

〈i,j〉,σc†i,σcj,σ + U

∑i

ni,↑ni,↓, (31)

where 〈i, j〉 denotes a sum over nearest neighbors.The electronic eigenstates fall into two bands for large

U at 1/2 filling - e.g., the occupied lower and unoccupiedupper Hubbard bands - separated by an energy U fordouble occupancies. Away from 1/2 filling, quasiparticles

develop. The microscopic Hamiltonians can be viewedas families of models related to the Hubbard model,such as the Falicov-Kimball model, Anderson model, orAnderson-Fano model, without loss of generality.

In essence, the sums over intermediate states inEq. (13) are separated into groups of bands lying far awayin energy from the initial and final states (i.e. the bandsprojected out) and bands lying nearby the initial andfinal states (considered bands with correlations). Theformer grouping of intermediate states are considered asin Section II.B to be approximated by the effective masscontribution, Eq. (24). The remaining terms involve ma-trix elements of the current operator between the remain-ing band of interest10.

The interaction of light with these downfolded elec-trons can be treated via the Peierls construction, in whichthe creation and annihilation operators develop a phase,

ci,σ → ci,σe−i e

c

ri−∞ A·d. (32)

The resulting scattering Hamiltonian obtained by ex-panding in powers of A reads

Hint =e

cj ·A +

e2

22c2

∑αβ

AαγαβAβ , (33)

where

jα(q) =∑k

∂ε(k)∂kα

c†σ(k + q/2)cσ(k − q/2), (34)

is a component of the current operator j and

γαβ(q) =∑k

∂2ε(k)∂kα∂kβ

c†σ(k + q/2)cσ(k − q/2) (35)

is the stress tensor operator. Both operators are thusformed from the energy dispersion of the downfoldedband structure. The matrix element can be written incompact form:

MF,I(q) =∑α,β

eiαe

sβM

α,β(q),

Mαβ(q) = 〈F |γα,β(q)| I〉 (36)

+∑

ν

⎛⎝⟨F∣∣∣jβ(qs)

∣∣∣ ν⟩⟨ν ∣∣∣jα(qi)∣∣∣ I⟩

Eν − EI − ωi

+

⟨F∣∣∣jα(qi)

∣∣∣ ν⟩⟨ν ∣∣∣jβ(qs)∣∣∣ I⟩

Eν − EI + ωs

⎞⎠ ,

10 We remark that this is not exact as it inaccurately treats resonantscattering processes occurring within the conduction band. Sincethese processes are usually taken to be damped, even though thisapproach is strictly speaking not exact, it provides a good start-ing point for considering Raman scattering in correlated single-band systems, and as such, has been widely used (Shastry andShraiman, 1991).

16

with the sum over intermediate states ν of the Hamil-tonian Eq. (30). The Raman cross section can be sepa-rated into non-resonant, mixed, and resonant contribu-tions:

R(Ω) = RN (Ω) +RM (Ω) +RR(Ω), (37)

where the nonresonant contribution is

RN (Ω) =∑I,F

exp(−βEI)Z γi,s

I,F γs,iF,I δ(EF − EI − Ω),

(38)the mixed contribution is

RM (Ω) =∑I,F,ν

exp(−βEI)Z

×[γi,s

I,F

(j(s)F,ν j

(i)ν,I

Eν − EI − ωi+

j(i)F,ν j

(s)ν,I

Eν − EI + ωs

)

+

(j(i)I,ν j

(s)ν,F

Eν − EI − ωi+

j(s)I,ν j

(i)ν,F

Eν − EI + ωs

)γs,i

F,I

]

× δ(EF − Eν − Ω), (39)

and the resonant contribution is

RR(Ω) =∑

I,F,νν′

exp(−βEI)Z

×(

j(i)I,ν j

(s)ν,F

Eν − EI − ωi+

j(s)I,ν j

(i)ν,F

Eν − EI + ωs

)

×(

j(s)F,ν′ j

(i)ν′,I

Eν′ − EI − ωi+

j(i)F,ν′ j

(s)ν′,I

Eν′ − EI + ωs

)

× δ(EF − EI − Ω). (40)

We have introduced the symbols

γi,s =∑αβ

eiαγαβ(q)es

β , j(i,s) =∑α

ei,sα jα(qi,s),

(41)and denote Oκ,λ as the matrix element 〈κ|O|λ〉. Some ofthe contributions are depicted diagrammatically in Fig-ures 8, 9, and 10 for the non-resonant, the mixed, andthe resonant terms, respectively. In the limit D → ∞these are the main diagrams to consider. Additional di-agrams involving multi-particle vertex renormalizationsgenerally contribute for finite dimensions (not shown).

In general, the matrix elements that enter intoEqs. (38–40) are not easy to calculate for an interactingsystem, so the summations are problematic to evaluate.In particular, one needs to evaluate the irreducible stressand current vertices, depicted in Figures 8-10 by thehatched symbols. Contributions to these vertex dressingsinclude many-particle renormalizations. A particularlycomplicated one is shown in Figure 11 which represents4-particle vertex corrections. Moreover, analytic contin-uation must be performed to obtain the Raman response

R(Ω) on the real axis from the imaginary axis. While thisis relatively straightforward for the non-resonant case,mixed and resonant cases are problematic because ofthe complicated dependences on each of the frequenciesωi,s,Ω which have to be analytically continued. Whilethe continuation has been worked out recently for thegeneral case, evaluating these diagrams for general inter-actions has proved elusive.

The overall complexity of the problem limits the evalu-ation of the light scattering cross section to generic inter-acting systems. Only recently, these diagrams have beenevaluated exactly in a DMFT treatment of the Falicov-Kimball model (Shvaika et al., 2004, 2005).

2. Correlated Insulators - Heisenberg Limit

To emphasize the generality of Eq. (13) we now con-sider the large U limit for the insulating half-filled two-dimensional Hubbard model. Following Shastry andShraiman (1991) we consider a system with N interactingelectrons in which the manifold of states can be classifiedby the number n of doubly occupied sites. The well-

FIG. 8 Feynman diagrams for non-resonant Raman scatter-ing. The wavy and the solid lines denote photon and electronpropagators, respectively. The cross-hatched rectangle is thereducible charge vertex. The symbol γ denotes the stress-tensor vertex of the corresponding electron-photon interac-tion. From Shvaika et al. (2005).

FIG. 9 Feynman diagrams for the mixed contributions to Ra-man scattering. The symbols jf and ji remind us to includethe relevant vertex factors from the current operator in theelectron-photon interaction. From Shvaika et al. (2005).

17

known Heisenberg Hamiltonian emerges from projectingthe Hubbard model down onto the reduced Hilbert space

FIG. 10 Feynman diagrams for the resonant contributions toRaman scattering. From Shvaika et al. (2005).

containing no double occupancies:

HHeisenberg = J∑i,δ

Si · Si+δ, (42)

with J = 4t2/U the Heisenberg exchange constant.Higher manifolds containing empty holes and doubly oc-cupied states can be labeled according to the net spinconfiguration σ of the N − 2n singly occupied sites, aswell as the locations R of the empty rhole and dou-bly occupied rdouble sites. We denote these states as|n; σ, R〉. These states are connected to each othervia

|1; σ′; R〉 = c†σ(rdouble)cσ(rhole) |0; σ〉, (43)

where |0; σ〉 = Πrc†σr

(r) |vac〉 and |vac〉 denotes thevacuum.

Light scattering thus occurs via transitions out of themanifold of singly-occupied states. To leading order forlarge U , only the n = 0 and n = 1 manifold of statescontributes to light scattering via Eq. (13), with n = 0denoting the ground state and n = 1 the manifold ofintermediate states having one doubly occupied and oneempty site. The first term containing mα,β cannot con-tribute for the half-filled lattice, and thus only interbandscattering between the upper and lower Hubbard bandsoccurs via the p ·A term. The energy difference betweenthese excitations is U to lowest order in t/U , allowing usto write the matrix element Eq. (13) in the form

FIG. 11 Feynman diagrams for a typical parquet-like renor-malization. This resonant diagram has a simultaneous hor-izontal and vertical renormalization by the two-particle re-ducible charge vertex. From Shvaika et al. (2005).

18

MF,I =∑

ν,r,r′,δ, δ′〈0, σI | js(r) es · δ | 1; σν;Rν〉〈1; σν;Rν | ei · δ′ji(r′) | 0, σF 〉

[1

U − ωi+

1U + ωs

], (44)

with the current operator defined as

ji,s(r) = i t[c†σ(r + δa · ei,s)cσ(r) − c†σ(r)cσ(r + δa · ei,s)]. (45)

Here δ is a unit vector connecting a site with its nearest neighbors. The intermediate states ν represent a sum overspin configurations and locations of both the doubly occupied and the hole sites. Substituting Eq. (43) into Eq.(44) collapses the intermediate state sum, leaving four terms connecting initial and final states. Using the identity1/2 − 2Si · Sj = c†(r + δa)c(r)c†(r)c(r + δa) valid in the manifold of singly occupied states, one obtains the lightscattering Hamiltonian of Elliot and Loudon (1963) and Fleury and Loudon (1968),

HEFL =∑r,δ

Sr · Sr+δa(es · δ)(ei · δ)[

1U − ωi

+1

U + ωs

]. (46)

We note that the polarization dependence is crucial aswell. For xx + yy polarizations projecting the fullysymmetric components, the light scattering Hamiltoniancommutes with the nearest neighbor Heisenberg Hamil-tonian and thus does not give inelastic scattering in theA1g channel. Moreover, B2g (xy) is also identically zero.As a result, a large signal appears only in the B1g chan-nel (xx − yy). These restrictions are lifted however iflonger range spin interactions are considered (Shastryand Shraiman, 1991).

The collapse of the intermediate states allowed us toreplace the operators with projected spin operators con-fined to the restricted Hilbert space of the n = 0, 1 man-ifolds. Thus, in this limited Hilbert space, the formalismis similar to non-interacting electrons in that the opera-tors appearing in the scattering matrix may be simplified.If the Hilbert space is enlarged to include larger mani-folds, then this would no longer be the case, and thusincluding terms to higher order in t/U becomes highlynon-trivial and is still one of the challenges to merge aweakly interacting picture into a strongly interacting one.

We note that Equation (46) was derived effectively asan expansion in t/(U − ωi). Therefore, the scatteringHamiltonian is limited to cases when both the numberof holes and double occupied sites are restricted and off-resonance conditions apply, with the incident photon en-ergy ωi far away from U . Efforts to extend the treat-ment to more general conditions involve understandingthe motion of holes or doubly occupied sites in an ar-bitrary spin background. This has proved to be a hardtask.

D. Electronic Charge Relaxation

In Sections II.B and II.C we reviewed the general for-malism of the theory of Raman scattering for weaklyand strongly correlated systems. In this subsection we

now specify the electronic states from which light can bescattered and review the various theoretical treatmentsfor specific models of interacting electrons. Emphasis isplaced upon how symmetry can be used to highlight elec-tron dynamics on regions of the BZ, and general featuresfor each model system will be presented. We first con-sider the case where the correlations among electrons areweak.

1. Weakly-Interacting Electrons

In this subsection, we consider electrons as having verywell defined eigenstates labeled by energy and momen-tum and having sharp spectral functions. Apart fromthe form of the energy dispersion ξk, the results arerather general and governed largely by phase space con-siderations. We must, however, consider the long-rangeCoulomb interaction in order to account for charge back-flow and screening, and we utilize the results derived inSection II.B and the general expression Eq. (29).

The use of χsc rather than χ takes into account themost drastic manifestation of the long-range Coulombinteraction, viz. screening. For weakly interacting elec-trons, the Random Phase Approximation (RPA) is ac-ceptable, which replaces χsc by the Lindhard functionfor non-interacting electrons. The response functions aredetermined by the Lindhard kernel,

χa,b(q,Ω) =2V

∑k

ak,qbk,qf(ξk) − f(ξk+q)

ξk − ξk+q + Ω − iδ, (47)

for general vertices a, b appearing in Eq. (29). In a freeelectron gas, the Raman response is given by Eq. (27)or equivalently, the last term in Eq. (29). For large q,collective excitations are unimportant and light scatter-ing occurs via creation of particle-hole excitations in theLandau continuum. However, as the only phase space

19

for creating particle-hole pairs comes from finite q trans-ferred from the photons, the resulting response is a con-tinuum varying linearly with Ω at small frequencies andextending up to a cut-off Ωc = vFq from the borders ofthe continuum (Mahan, 2000). The low-energy intensityis proportional to q2, and the only excitation left at q = 0is the collective plasmon. The Raman response for thefree electron gas is shown in Figure 12.

For electrons in a solid, however, the non-parabolicityof the energy dispersion results in charge fluctuationswhich are anisotropic in the small q limit and thus cansurvive screening and give more weight at low energytransfers. Formally, an additional contribution to the re-sponse is given by the first two terms in Eq. (29). Yetphase space restrictions still produce an inescapable cut-off at Ωc = vF q (Wolff, 1968), and the response resemblesthat shown in Figure 12. This is also the case if scatter-ing occurs for the 2 dimensional electron-gas (2DEG) inthe absence of a magnetic field (Jain and Das Sarma,1987; Mishchenko, 1999) or for complex Fermi surfaces(Ipatova et al., 1983). An RPA treatment for resonantscattering has been given by Wang and Das Sarma (1999,2002).

Recently, substantial progress has been made in under-standing the Raman response in the integer or fractionalquantum Hall regimes of the 2DEG. Space limitations donot allow us to review these systems; so for brevity, wecite only a recent reference (Richards, 2000).

2. Impurities

Excitations at low energies in non-interacting elec-tronic systems can arise for small q via electronic scat-tering from impurities, where momentum contributed byimpurity scattering can provide phase space for electron-hole creation which is anisotropic in the Brillouin zone.For example, if one considers a general electron-impurity

FIG. 12 Raman response of an electron gas. From Platzman(1965).

interaction of the form

Himp =∑

k,k′,σ

Vk,k′c†k,σck′,σ, (48)

with an anisotropic interaction Vk,k′ , the gauge invari-ant Raman response is given via the diagrams presentedin Figure 8. If we make a symmetry decomposition ofthe scattering amplitude γ(k) =

∑L γLΦL(k) in terms

of basis functions ΦL of the Brillouin zone, the result-ing response in channel L corresponding to a particu-lar light polarization orientation has a Drude Lorentzianform (Devereaux, 1992; Fal’kovskiı, 1989; Zawadowskiand Cardona, 1990):

χ′′(q,Ω) = NFγ2L

Ωτ∗L1 + (Ωτ∗L)2

, (49)

with NF the density of states at the Fermi level. 1/τ∗L =1/τ +Dq2 − 1/τL is the effective scattering rate where

1/τ = 1/τL=0 = niNF

∫dSk

S

∫dSk′

S| Vk,k′ |2, (50)

involving an integration of the Fermi surface Sk normal-ized to the Fermi area S. Here ni is the impurity con-centration, D = 1

3v2Fτ is the diffusion constant. The

anisotropy of the impurity scattering is characterized viaorthonormal basis functions ΦL

Vk,k′ =∑L,L′

Φ∗L(k)ΦL(k′)VL,L′ , (51)

using an intelligent basis where the interaction is diagonalVL,L′ = δL,L′VL. Then, 1/τL = 2πniNFVL and 1/τL=0

is the dominant contribution.The resulting Raman spectrum (Figure 13) grows lin-

early with frequency Ω, decays as 1/Ω, and has a peakwhen Ωτ∗L equals 1. The width of the Lorentzian re-flects the rate for which charge density excitations hav-ing symmetry L decay into all other channels. The lightpolarizations select the type of excitation L created, andthus allow a way to probe the anisotropy of the impurity-electron interaction. The decay of the charge density

FIG. 13 Raman response from impurity scattering in an oth-erwise non-interacting system. From Zawadowski and Car-dona (1990).

20

fluctuations can occur via finite q through the diffusionterm and all contributions other than VL which relax elec-trons out of the state L. Put another way, Dq2 +1/τ are“scattering out” processes, while 1/τL is a “scattering in”process, giving an effective scattering rate 1/τ∗L. This isa consequence of a gauge invariant treatment including

21

tions in the Hubbard (Freericks et al., 2001, 2003a) andFalicov-Kimball models (Freericks and Devereaux, 2001).

Two aspects of the Raman response are generally inthe main focus: the frequency dependence of the broadcontinuum extending well past qvF, and the polarizationdependence. We discuss first the spectral response.

In the context of the cuprates, Varma and cowork-ers pointed out that a flat, nearly frequency indepen-dent response could be obtained if the imaginary part ofthe electron self-energy depended linearly on frequency(Varma et al., 1989a; Varma, 1989b). The response isthen given in terms of the scale-invariant ratio of ω/kBTand approaches a constant at large frequency transfers.This can be understood phenomenologically by replac-ing 1/τ∗L with 1/τ∗L(Ω, T ) ∝ max(kBT, Ω) in Eq. (49).11A scale-invariant response at low frequencies is a gen-eral consequence of systems in proximity to a quantum-critical point, but this scale invariance is broken outsidethe quantum critical regime. “Marginal Fermi liquid”behavior emerges for instance when scattering is consid-ered in a nested Fermi liquid (Virosztek and Ruvalds,1991, 1992), low Fermi-energy systems (Dahm et al.,1999; Devereaux and Kampf, 1999), and slave-boson sys-tems (Bang, 1993). A broad background very similarto “marginal” behavior is also found for strongly coupledelectron-phonon systems (Itai, 1992; Kostur, 1991, 1992).As a representative example, we show the response calcu-lated by Virosztek and Ruvalds (1992) for a nested Fermiliquid in Figure 14.

Low energy electron dynamics can be extracted by

FIG. 14 Raman response as a function of temperature ob-tained by Virosztek and Ruvalds (1992) for a system with anested Fermi surface. The upturn at low frequencies is to the(log(Ω)) frequency dependence of the effective mass in thismodel.

11 This is only an approximation since 1/τ∗L(Ω, T ) depends now on

energy. Causality requires that the relaxation function has realand imaginary part, M(Ω, T ) = Ωλ + i/τ∗

L (Gotze and Wolfle,1972; Opel et al., 2000).

studying the Raman response in the limit Ω → 0. Ne-glecting vertex corrections, the low frequency responsereads (Devereaux and Kampf, 1999; Venturini et al.,2002a)

χ′′µ( Ω → 0) = ΩNF ×

×⟨γ2

µ(k)∫dξ(−∂f0

∂ξ

)Z2

k(ξ, T )

2Σ′′k(ξ, T )

⟩. (54)

Here, NF is the density of electronic levels at theFermi energy EF, Σ′′

k is the imaginary part of thesingle-particle self energy related to the electron life-time as /2Σ′′

k(ω, T ) = τk(ω, T ), Zk(ω, T ) = (1 −∂Σ′

k(ω, T )/∂ω)−1 is the quasiparticle residue, f0 is theequilibrium Fermi distribution function, and 〈· · ·〉 de-notes an average over the Fermi surface. Thus the inverseof the Raman slope

Γµ(T ) =[∂χ′′

µ(Ω)∂Ω

]−1

(55)

measures the effective “scattering rate” of the quasipar-ticles in a correlated metal, and can be best thought ofas a “Raman resistivity”.

In systems with isotropic interactions, the polarizationdependence drops out and the slope of the low-frequencyRaman response is given in terms the low energy quasi-particle scattering lifetime, Γµ(T ) ∝ /τ(T ), as an ex-tension of Eq. (49). Yet, in strongly correlated systems,the quasiparticle residue Z and, importantly, vertex cor-rections, enter as well. In a correlated or a strongly disor-dered metal (near an Anderson transition, e.g.), however,a finite energy might be necessary to move an electronfrom one site to another one. Thus, in spite of a non-vanishing density of states at the Fermi level, as observedin an ARPES experiment for instance, no current can be

FIG. 15 Inverse Raman slope (see Eq. (55)) close to a metal-insulator transition at a value U =

√2 in the Falicov-Kimball

model for D = ∞ (Freericks and Devereaux, 2001). All ener-gies and temperatures are measured in terms of the hoppingt.

22

transported and Γµ(T ) /τ(T ). This is an importantdifference between single- and two-particle properties.

Figure 15 displays the inverse Raman slope defined inEq. (55), as determined via a DMFT treatment of theFalicov-Kimball model in the vicinity of a metal-insulatortransition, as a function of the Coulomb repulsion U(Freericks and Devereaux, 2001). It provides an illus-trative example of how Fermi-liquid-like features evolveas the lifetime of putative quasiparticles increases due todecreased role of correlations. For small U , the corre-lated metal displays an inverse slope ∝ T 2 as a canonicalFermi liquid in the metallic state. A pseudogap open-ing in the density of states with increasing U drives theinverse slope into insulating behavior, increasing as thetemperature decreases.

As a second important application Eq. (55) can illu-minate the anisotropy of electron dynamics due to themomentum-dependent weighting factors of the polariza-tion orientations and self energies. The geometry of lightscattering orientations, as given by the form factors listedin Table I, project out the ratio of quasiparticle residuesand scattering rates in different regions of the BZ. As aconsequence, the Raman spectra show polarization de-pendent behavior determined largely by the self energiesand vertex corrections near the regions projected by thescattering vertices γ. Figure 16 plots the B1g and B2g

Raman response calculated in a spin-fermion model inwhich electron scattering is most pronounced involvingantiferromagnetic reciprocal lattice momentum transfersQ = (π, π), leading to “hot” quasiparticles near the BZaxes (projected by B1g form factors) and “cold” quasi-particles along the BZ diagonals (projected by B2g formfactors). Therefore, the Raman response has a sharpquasiparticle peak for B2g scattering at low energies dueto the long quasiparticle lifetimes, while the response inB1g is dominated by strong incoherent scattering leadingto a suppression of the quasiparticle peak at low energiesand an essentially structureless continuum.

Lastly, we note that the non-resonant Raman re-sponse has also been calculated for exchange of fluctu-ation modes at wavevectors Q and -Q for systems near a

FIG. 16 Polarization-dependent Raman response in a spin-fermion model for a fixed temperature for three different val-ues of the coupling constant. From Devereaux and Kampf(1999).

spin-density wave instability (Brenig and Monien, 1992;Kampf and Brenig, 1992; Venturini et al., 2000) and acharge-density wave instability (Caprara et al., 2005).Here, the Raman response is sensitive to the light po-larizations and has a peak centered at twice the energyof the fluctuating mode.

4. Interacting Electrons - Resonant Response

In addition to the non-resonant response, one has themixed and resonant contributions to consider. Typi-cally these diagrams are neglected in the weak correlationlimit, as they can be summed into two-particle responsefunctions as discussed in Section II.B. In the insulatingcase, only the resonant terms are kept, as the studies fo-cus on excitations across a charge-transfer or Hubbardgap. This has been calculated in systems exhibiting 1D-Luttinger behavior (Kramer and Sassetti, 2000; Sassettiand Kramer, 1998; Sassetti et al., 1999; Wang et al., 2004)where bosonization techniques can be applied. Yet, gen-erally treating all diagrams on equal footing is technicallydemanding. Only recently, an exact evaluation in DMFThas been performed (Shvaika et al., 2004, 2005).

It is well known that many of the Raman signals incorrelated metals and insulators display complicated de-pendences on incoming photon frequency ωi. For exam-ple, the B1g two-magnon feature at roughly 350 meV inthe thoroughly studied insulating parent cuprates has aresonance for incident photon energies near 3 eV. As a re-action to the experimental results in the cuprates, muchtheoretical work has been devoted to Raman scatteringin a two-dimensional Heisenberg antiferromagnet usingthe Fleury-Loudon model Eq. (46).

In the nearest neighbor Heisenberg antiferromagnet,one treats the spin operators using a Dyson-Maleev rep-resentation of magnons with dispersion 2J . Due to theB1g form factor, in the absence of magnon-magnon inter-actions at T=0, a sharp peak at 4J appears as the top ofthe magnon dispersion is projected out, and the responseabruptly drops to zero for large Raman shifts (Sandviket al., 1998). However, since the light scattering is local-ized to neighboring spins, magnon-magnon interactionsmust be included, and the peak becomes more symmetricand shifts to ∼ 3J via breaking exchange bonds betweenlocal neighbors, as shown in Figure 17.

There have been many developments on the Fleury-Loudon model, which has been addressed via criticalfluctuation analysis (Halley, 1978), series expansions(Singh et al., 1989), lower (Morr and Chubukov, 1997;Parkinson, 1969) and higher (Canali and Girvin, 1992;Chubukov and Frenkel, 1995b) order spin wave theories,t−J studies at finite doping (Prelovsek and Jaklic, 1998),exact diagonalizations of small clusters (Tohyama et al.,2002), excitonic cluster approaches (Hanamura et al.,2000), finite temperature Quantum Monte Carlo meth-ods (Sandvik et al., 1998), and studies of bilayer effects(Morr et al., 1996), 2-leg spin ladders (Jurecka et al.,

23

2001) and ring exchange (Katanin and Kampf, 2003),giving a thorough treatment of two magnon scatteringfrom spin degrees of freedom in the non-resonant regime.Scattering from channels other than B1g, and describingthe anisotropic lineshape of the response, have been ad-dressed via longer range spin exchange interactions andby exact diagonalizations of magnons coupled to phonons(Freitas and Singh, 2000), using an earlier approach ofLorenzana and Sawatzky (1995). Lastly, recent develop-ments concern scattering from orbiton degrees of freedom(Okamoto et al., 2002) and scattering within a resonant

INITIAL STATE INTER. STATE FINAL STATE

FIG. 17 Cartoon of the two-magnon scattering process in a2D Heisenberg antiferromagnet. An incident photon causesan electron with spin σ to hop leaving a hole and creating adouble occupancy in the intermediate state with energy ∝ U .One particle of the double with spin −σ hops back to thehole site liberating a photon with energy ∼ U − zJσ, leavingbehind a locally disturbed antiferromagnet with z exchangebonds broken in the final state as indicated by the dottedlines.

FIG.18Thedependenceof(a)two-magnonB1gR a m a n i n - t e n s i t y ( s h o w n i n t h e i n s e t f o r ffi= 8 t) a n d ( b ) a b s o r p t i o n s p e c t r u m o n t h e i n c o m i n g p h o t o n e n e r g y ffiina20-siteclus-t e r o f t h e H u b b a r d m o d e l w i t h U= 1 0 t(J= 0 .4t).Thesolid

linein(b)isobtainedbyperformingaLorentzianbroadening

w i t h a w i d t h o f 0 .4tonthedeltafunctionsdenotedbyverticalbars.FromTohyamaetal.(2002).valencebondpicture(Hoetal., 2 0 0 1 ) . 12Theseapproachesfailwhenthelaserfrequencyistunedn e a r a n o p t i c a l t r a n s i t i o n . I n t h i s r e g i m e , b a s e d o n

aspindensitywaveapproach,ChubukovandFrenkel

( 1 9 9 5 a , b ) h a v e f o r m u l a t e d a s o c a l l e d “ t r i p l e - r e s o n a n c e ”

t h e o r y f r o m w h i c h i m p o r t a n t features of the sp ectra canbederived.UsingaSDWapproachtotheHubbard

model,theyfoundthatadditionalresonantdiagramsof

thetypeshowninFigure10(h)-(j)contributetothe

usua l L o udo n-Fleur y ter ms, a nd der ived a r eso na nt pr o -

fi l e i n g o o d a g r e e m e n t w i t h e x p e r i m e n t s . I n a d d i t i o n , r e -

c e n t r e s u l t s b y T o h y a m a etal.(2002) have been obtained

fortheRamanresponseintheresonantlimitfromboth

s p i n a n d c h a r g e d e g r e e s o f f r eedom.InFigure18weshowt h e i r r e s u l t s f r o m e x a c t d i a g o n a l i z a t i o n o f t h e H u b b a r d

m o d e l w i t h a 2 0 - s i t e c l u s t e r ( T o h y a m a etal., 2 0 0 2 ) . T h e t w o - m a g n o n r e s p o n s e a t r o u g h l y ff∆ = 2 .7Jis resonantly

e n h a n c e d w h e n t h e i n c i d e n t photonfrequencyistunedtot h e M o t t g a p s c a l e U,inqualitativeagreementwiththe

r e s u l t s o f C h u b u k o v a n d F renkel (1995a,b). B oth a p-p r o a c h e s p r e d i c t a r e s o n a n t p r o fi l e f o r t w o - m a g n o n R a -

manwhichdiffersfromtheabsorptionprofile,asshown

i n F i g u r e 1 8 9 T o h y a m a etal.( 2 0 0 2 ) h a v e p o i n t e d o u t

t h a t t h e r e s o n a n c e e n e r g i e s forthe absorption spectruma n d t h e t w o - m a g n o n r e s p o n s e a r e n o t t h e s a m e , d u e t o

d i ff e r e n c e s i n S D W c o h e r e n c e f a c t o r s . 59InteractingElectrons-FullResponseA n a p p r o a c h t r e a t i n g t h e f u l l f e r m i o n i c d e g r e e s o f f r e e -

d o m a n d , s i m u l t a n e o u s l y , t r e a t i n g n o n - r e s o n a n t , m i x e d ,

a n d r e s o n a n t s c a t t e r i n g o n e q u a l f o o t i n g , i s s t i l l i n i t s i n -

fancy.Thetheoreticalchallengeincalculatingthefullin-

elasticlightscatteringresponsefunctionisthatthemixed

diagramsinvolvethree-particlesusceptibilitiesandthe

resonantdiagramsinvolvefour-particlesusceptibilities.

Onlyintheinfinite-dimensionallimit,wheremostofthe

m a n y - p a r t i c l e v e r t e x r e n o r m a lizationsvanish(allthree-particleandfour-particleverticesdonotcontribute;only

t h e t w o - p a r t i c l e v e r t i c e s e n t e r),onecanimaginearrivinga t e x a c t r e s u l t s . A s a n e x c e p t i o n , t h e f u l l R a m a n r e -

sp o nse functio n ca n b e ca lcula ted in the Fa licov -K imba ll

model,becausethetwo-particleirreduciblechargevertex

isknownexactlyinthelimitoflargedimensions(Freer-

i c k s a n d M i l l e r , 2 0 0 0 ; S h v a i k a , 2 0 0 0 ) .

Recently,Shvaikaetal.( 2 0 0 4 , 2 0 0 5 ) o b t a i n e d t h e

fullelectronicRamanresponsefunction,includingcon-

t r i b u t i o n s f r o m t h e n o n - r e s o n a n t , m i x e d , a n d r e s o n a n t

p r o c e s s e s w i t h i n a s i n g l e - b a n d m o d e l . I n g e n e r a l , t h e

resonanceeffectscancreateordersofmagnitudeenhance-

m e n t o v e r t h e n o n - r e s o n a n t r e s p o n s e , e s p e c i a l l y w h e n t h e

12ThereaderisalsoreferredtothereviewarticlebyLemmense t al.(2003)forreviewsonthetheoreticaltreatmentsofmagneticlightscatteringinlow-dimensionalquantumspinsystems.

24

A

res

pons

e

B

res

pons

e

B

res

pons

e1g

1g2g

Frequency [t]

0.4

0.3

0.2

0.1

0

0.08

0.06

0.04

0.02

00.50.40.30.20.1 0

0 0.4 0.8 1.2 1.6 2

(a)

(b)

(c)

FIG. 19 Raman response in the 1/2-filled Falicov-Kimballmodel (U = 2t) as a function of transfered frequency for vari-ous temperatures for fixed incident photon frequency ωi = 2t.The thickest curve is T/t = 0.05, and the temperature in-creases to 0.2, 0.5, and 1 as the curves are made thinner.From Shvaika et al. (2004).

incident photon frequency is slightly larger than the fre-quency of the non-resonant feature. The resulting Ramanresponse is a complicated function of the correlations, thetemperature, the incident photon energy, and the trans-fered energy. It was found that resonance effects are dif-ferent in different scattering geometries, corresponding todifferent symmetries of the charge excitations scatteredby the light.

Resonance effects were found as a function of both theincoming and the outgoing photon frequencies ωi,s. Adouble resonance - occurring when the energy denomi-nators of two pairs of the Greens functions, appearingin the bare response shown in Figure 10, approach zero- gives the strongest resonant enhancement of the re-sponse(Shvaika et al., 2004). In addition, an interestingresonance effect on both the charge-transfer peak andthe low-energy peak was found when the incident pho-ton frequency is of the order of the interaction strength,showing that in general the total response cannot be welldescribed as a uniform resonance enhancement of the sep-arable non-resonant response. In agreement with the re-sults of Tohyama et al. (2002), for an antiferromagneticsystem this is a direct consequence of the inseparabilityof energy scales in the correlated electron problem, in

contrast to non-interacting electrons.Shown in Figure 19 is the temperature and symmetry-

dependent Raman response, including non-resonant, res-onant, and mixed terms in the Falikov-Kimball model.In the insulating phase, spectral weight is depleted forsmall energy transfers and piles up into the excitationsat energies of order U as the temperature is lowered. Thetransfer of spectral weight from lower to higher energiesoccurs across a temperature independent so-called isos-bestic point. An isosbestic point also appears in studiesof the Hubbard model (Freericks et al., 2001, 2003a), im-plying that it is a generic feature of the insulating phase,regardless of the microscopic origin of the phase. Wenote that isosbestic behavior already appears in the non-resonant contributions for B1g scattering. In the A1g

and B2g symmetries, it emerges only if resonant termsare included.

The local treatment of self energies in the single-siteDMFT approach imposes limitations on the theory oflight scattering in correlated systems. In particular, thefull polarization dependence of the Raman spectra woulduncover the way in which correlations affect electron dy-namics in regions of the BZ, providing a two-particlecomplement to ARPES, for example. Progress here mostlikely will come from cluster dynamical mean field the-ory able to treat nonlocal and anisotropic interactions ina coarse-grained manner.

6. Superconductivity

As discussed in Section II.D.1, in the absence of inter-actions, there is no phase-space for low energy Ramanscattering for q = 0 momentum transfers. In the super-conducting state, phase space restrictions are lifted sincelight can break q = 0 Cooper pairs if the energy of thelight is greater than 2∆. Thus the Raman response be-comes non-trivial, yet easily formulated, in BCS theory.As a consequence, there has been an enormous amountof theoretical work devoted to light scattering for tem-peratures below Tc as an extension of the theory for non-interacting electrons in the normal state. We review thatwork here.

In the superconducting state, focus has been tradi-tionally placed on the two-particle non-resonant responsein BCS theory. Formally the Raman response is givenby generalizing Eqs. (52-53) in particle-hole space usingPauli matrices τi=0..3 in Nambu notation (Nambu, 1960):

χ(q = 0, iΩ) = − 2V β

∑iω

∑k

Tr[γ(k)G(k, iω)Γ(k; iω; iΩ)G(k, iω + iΩ)

], (56)

25

where Tr denotes the trace, and

Γ(k; iω; iΩ) = γ(k) +1V β

∑iω′

∑k′Vi(k − k′, iω − iω′)τiG(k′, iω′)Γ(k′; iω′; iΩ)G(k′, iω′ + iΩ)τi. (57)

Here the bare Raman vertex of coupling to charge isγ = τ3γ and the interaction Vi determines the channel ofthe vertex corrections. For example Vi=3 corresponds tointeractions coupling electronic charge, while Vi=0 corre-sponds to spin interactions.

For the case of weak correlations, the Green’s functionsappearing in Eqs. (56) and (57) are given by the BCSexpression

G(k, iω) =iωτ0 + ε(k)τ3 + ∆(k)τ1

(iω)2 − E2(k), (58)

with E2(k) = ξ2(k) + ∆2(k) the quasiparticle energies.In the weak coupling limit for the BCS approximation,Vi=3 = −V for phonon-mediated pairing.

By far, the superconducting state has received thelargest amount of attention from theory, starting fromthe seminal contribution of Abrikosov and Fal’kovskiı(1961), which predated the observation of the effect by19 years. The main focus in the early years was to studythe 2∆ features in conventional s−wave superconduc-tors with small and large coherence lengths (Abrikosovand Fal’kovskiı, 1961, 1987; Abrikosov and Falkovsky,1988; Klein and Dierker, 1984), including the effects ofCoulomb screening (Abrikosov and Genkin, 1973) andexamining the temperature dependence (Tilley, 1972).If one neglects vertex corrections, the q-dependent Ra-man response in a superconductor is given by a projectedMaki-Tsuneto function (Maki and Tsuneto, 1962),

χ′′a,b(q,Ω + iδ) =

1N

∑k

ak,qbk,q

×A+(k,q) [f(E(k)) − f(E(k + q))]

(1

Ω + iδ − E(k) + E(k + q)− 1

Ω + iδ + E(k) − E(k + q)

)

+A−(k,q) [1 − f(E(k)) − f(E(k + q))](

1Ω + iδ + E(k) + E(k + q)

− 1Ω + iδ − E(k) − E(k + q)

), (59)

with the coherence factors A±(k,q) = 1± ξ(k)ξ(k+q)−∆(k)∆(k+q)E(k)E(k+q) . A more common expression is the Raman response

for q = 0 which simplifies to

χ′′a,b(q = 0,Ω + iδ) =

2N

∑k

akbk

[ | ∆(k) |E(k)

]2tanh

(E(k)2T

)(1

2E(k) + Ω + iδ+

12E(k) − Ω − iδ

), (60)

The full Raman response, including charge screening, isonce again given by Eq. (29), in which the vertices a, bare replaced by the Raman (a, b = γ) and pure charge(a, b = 1) vertices (Abrikosov and Genkin, 1973).

For the case of an isotropic gap (∆(k) = ∆) and mo-mentum transfers q = 0, a threshold and a square-rootdiscontinuity appears at twice the gap edge ∆, reflectingthe two-particle density of states. For finite q, the singu-larity is cut-off due to breaking Cooper pairs with finitemomentum, and the peak is shifted out to frequencies ofroughly vFq as in the normal state (Figure 20). Qualita-tively similar behavior is obtained for disordered s−wavesuperconductors (Devereaux, 1992) in which 1/τ∗L (seeEq. (49)) assumes the role of vFq .

Further advances in the theory for conventionals−wave superconductors were made for energy gaps with

small anisotropy (Klein and Dierker, 1984), coexistencewith charge-density wave order (Balseiro and Falicov,1980; Littlewood and Varma, 1981, 1982; Tutto and Za-wadowski, 1992), layered superconductors (Abrikosov,1991), impurities (Devereaux, 1992, 1993), and final-stateinteractions (Devereaux, 1993; Devereaux and Einzel,1995a; Klein and Dierker, 1984; Monien and Zawadowski,1990).

We note in particular that the variation with k of theRaman vertices γ(k) is coupled to the k-dependence ofthe energy gap ∆(k) (see, e.g., Eq. (60)), leading toa strong polarization dependence of the spectra. Forisotropic s-wave superconductors, the vertex does not af-fect the lineshape, and thus the spectrum is polarizationindependent, apart from an overall prefactor. For thiscase, a polarization dependence can be generated in BCS

26

theory by taking into account channel-dependent final-state interactions (Bardasis and Schrieffer, 1961) and/orimpurity scattering. However, for the most part this onlyproduces a channel dependence in the vicinity of the gapedge, and thus the main feature of the response is theuniform gap existing for all polarizations. For anisotropicenergy gaps, the symmetry dependence of the spectra isa direct consequence of the k-summation (angular aver-aging), which couples gap and Raman vertex and leads toconstructive (destructive) interference if the vertex andthe gap have the same (different) symmetry.

Generally, in superconductors with nodes of the energygap, power-laws in the low frequency and/or tempera-ture variation of transport and thermodynamic quanti-ties emerge, replacing threshold or Arrhenius behaviorubiquitous in isotropic superconductors. However, dueto the averaging over the entire Fermi surface, the power-laws themselves do not uniquely identify the ground statesymmetry of the order parameter, but only can give thetopology of the gap nodes along the Fermi surface, e.g.,whether the gap vanishes on points and/or lines. Thus,one cannot distinguish between different representations

27

For dx2−y2 superconductors, the interplay of polariza-tions and gap anisotropy can be simply drawn. Refer-ring to Figure 7, B1g orientations project out excitationsaround the principle directions (M -points or anti-nodalregions) of the BZ where the superconducting gap is max-imal and where the van Hove singularity is located, whileB2g orientations project the nodal regions along the di-agonals. As a consequence, the Raman response has apeak at 2∆max for B1g and at slightly lower energy forB2g. The polarization dependence also enters the low fre-quency behavior. Since line nodes yield a linear depen-dence on energy of the density of states, the B2g responsedepends linearly on Ω in the limit Ω → 0 for a gap van-ishing on the diagonals. For B1g orientations, in contrast,the Raman vertex vanishes along with the energy gap atthe same points in the BZ. This yields an additional Ω2

contribution from the line nodes of the vertex, and theresulting response varies as Ω3. The unscreened A1g re-sponse measures an overall average throughout the BZand thus picks up the gap maxima as well as the lineardensity of states. This is shown quantitatively in Fig-ure 21. The frequency power laws also translate in lowtemperature power laws of the response in the small fre-quency limit (Devereaux and Einzel, 1995a).

Disorder effects generally smear peak features at largerenergies and change the B1g exponent to 1, similar tothe change in the low temperature NMR rate for d-wavesuperconductors (Devereaux, 1995b). Moreover, simi-larities between the in-plane conductivity and B2g Ra-man follows from the BZ weighting around the nodes,while the B1g response is qualitatively similar to the c-axis conductivity due to the weighting around the anti-nodes (Devereaux, 2003b). For example the residual in-plane conductivity as T → 0 is universal and given byσ(T = 0) = ne2/mπ∆0, the slope of the B2g response2NF/π∆0 is also universal and insensitive to impurityeffects, while the B1g channel and c-axis conductivityare non-universal, having additional impurity dependentprefactors (Devereaux, 1995b; Devereaux and Kampf,1997; Devereaux, 2003b). The slope of the B2g responsefollows the temperature dependence of the in-plane con-ductivity, and both possess a peak at intermediate tem-peratures due to a balance of DOS and lifetime effectsas temperatures are lowered from Tc. Yet, both the out-of-plane conductivity and the B1g response do not showa peak due to the more rapid variation of the projectedDOS coming from antinodal portions of the BZ.

We note that for a dxy energy gap, the above discussionapplies accordingly, with the role of B1g and B2g sym-metry reversed. It was indeed an important developmentto show that Raman scattering is unique in determiningtwo-particle electron dynamics independently in differentregions of the BZ in the superconducting state.

While the low frequency power-laws are insensitive toband structure (such as the shape of the Fermi surface),the polarization selection rules can select features of theband structure at higher energies. For example, in thecuprates, the van Hove singularities from (π, 0) and re-

lated points yields a peak at twice the quasiparticle en-ergy E(k) in A1g and B1g channels, as shown in Figure(21). For multiple bands, including the case of severalFermi surface sheets, the responses for crossed polariza-tions are simply additive, yet for A1g channels due tobackflow effects an additional interference term can bepresent if the charge fluctuations are different for the dif-ferent sheets (Devereaux et al., 1996; Krantz and Car-dona, 1994).

Quite generally the backflow yields substantial reor-ganization of spectral weight around 2∆max comparedto the bare response(Branch and Carbotte, 1995; Dahmet al., 1999). This is because the term χγ,1, which con-tributes in channels having the symmetry of the lattice, ispeaked and large at the same position as the unscreenedterm χγ,γ. Not surprisingly, the reorganization dependsdelicately on the relative momentum dependences of theRaman vertices and energy gap (number of BZ harmonicsfor example, as shown in Figure 22), as well as on detailsof the band structure (Branch and Carbotte, 1996; Dev-ereaux et al., 1994a; Krantz and Cardona, 1994; Strohmand Cardona, 1997).

For the cuprates, Raman vertices have been calculatedusing LDA (Strohm and Cardona, 1997), but limitedprogress has been made in including the contributions ofsubstantial electronic correlations. In most other cases,either the effective mass approximation has been used incalculations or simply a symmetry classification has beenmade. While a detailed lineshape analysis can be appliedbased purely on symmetry as explained above, it must bekept in mind that even a comparison of overall intensitiesbetween different geometries can at best be qualitative.

Yet, the continuation of these treatments from the su-

FIG. 22 Comparison of the screened A1g response ob-tained for different number of dx2−y2 gap harmonics ∆(k) =

∆0[cos(kxa) − cos(kya)]/2 + ∆1[cos(kxa) − cos(kya)]3/8+∆2[cos(kxa) − cos(kya)]5/32. Here a − d correspond to theset ∆1,2 = (0, 0), (1, 0), (0, 1), and (1, 1), respectively, and ∆0

has been rescaled to give the same value for the maximumgap. Generally the large peak at 2∆ (as shown in Figure 21)is suppressed by the backflow terms. From Devereaux et al.(1996).

28

perconducting to the normal state is not straightforward.As can be seen directly from Eq. (60), the intensity van-ishes proportional to ∆2 as T approaches Tc. Hence, toavoid phase space limitations in the absence of Cooperpairs, an additional source for electronic scattering, suchas the one mediating the formation of Cooper pairs, mustbe included. While strong coupling extensions of Ramanscattering in d−wave superconductors have recently beenpresented (Dahm et al., 1999; Devereaux and Kampf,2000; Jiang and Carbotte, 1996), a merging of the nor-mal and the superconducting states is poorly understood.This would require a not yet existing microscopic descrip-tion of the formation of d−wave superconductivity fromthe normal state.

7. Collective Modes

Raman scattering has the almost unique ability to sortout collective modes of the two-particle response in dif-ferent symmetry channels, owing to the freedom to inde-pendently adjust the two polarization vectors. The col-lective mode spectrum one obtains depends upon whichinteractions are included in Eq. (57). We first discussthe general consequences based on gauge-invariance andfocus on exciton-like modes.

In order to form a fully gauge-invariant theory, the in-teractions responsible for superconductivity appear notonly in G, but must also be included as vertex renormal-izations Γ. In this way, the Raman response from purecharge density fluctuations in the superconducting stateyields the Goldstone mode from the broken gauge sym-metry - the phase or Anderson-Bogoliubov mode (An-derson, 1958; Bogoliubov et al., 1959; Nambu, 1960). Inthe absence of the long-range Coulomb interaction, thismode is a soft sound mode, yet the Coulomb interac-tions - inescapable for q=0 - push the sound mode upto the plasma frequency via the Higgs mechanism. Asa result, particle-number conservation is satisfied in thesuperconducting state and χsc(q = 0,Ω) = 0, indepen-dent of whether one considers Bloch states (Abrikosovand Genkin, 1973; Klein and Dierker, 1984; Monien andZawadowski, 1990) or Anderson exact eigenstates of thedisordered problem (Devereaux, 1993).

However, additional modes of excitonic origin may ap-pear if one considers further interactions between elec-trons in clean (Bardasis and Schrieffer, 1961) and dis-ordered (Fulde and Strassler, 1965; Maki and Tsuneto,1962) conventional superconductors. These excitons ap-pear split off from the continuum at Ω < 2∆ if theinteraction occurs in higher momentum channels orthog-onal to the BCS condensate.

Since Raman scattering couples to anisotropic chargedensity fluctuations with symmetry selectivity to differ-ent channels L, the light polarizations can be used todetermine the exact nature of bound states. Balseiroand Falicov (1980) considered the formation of a phonon-Cooper pair bound state due to electron-phonon coupling

though neglecting channels higher than L = 0. How-ever, this mode is canceled by the backflow applyinggenerically to all systems. Finite L exciton formationin clean and disordered superconductors, and the result-ing appearance in Raman scattering, have been consid-ered explicitly by Monien and Zawadowski (1990) andDevereaux (1993), respectively, bringing the symmetryof the exciton and the polarization dependence to light.

We show in Section III.B below that the effect of fi-nal state interactions can be substantial in strongly cou-pled conventional superconductors. This demonstratesthe strength of the electron-phonon coupling, not onlyin general, but also specifically in channels orthogonalto the ground state. Interestingly, the lattice instabilityfound in some of these materials has the same symme-try as the collective mode and the electronic states whichapparently drive the transition (Weber, 1984).

For dx2−y2 superconductors, the collective mode spec-tra have been investigated thoroughly by Devereaux andEinzel (1995a) and others (Dahm et al., 1998; Manskeet al., 1997, 1998; Strohm et al., 1998a; Wu and Griffin,1995a,b). It was shown that the Anderson-Bogoliubovmode appears in A1g channels and massive modes canappear in other channels. Since the pair state hasonly one representation in the D4h group, massive col-lective modes arise when one considers interactions inorthogonal channels. Recently, it has been suggestedthat the presence of collective modes may allow oneto distinguish charge- or spin-mediated d−wave pairing(Chubukov et al., 1999, 2006), highlighting the possibleimportance in the context of the cuprates.

Generally, the collective mode spectrum can be quitediverse in unconventional superconductors. In principle,additional broken continuous symmetries can exist, suchas SOS

3 spin rotational symmetry in spin-triplet systemsand SOL

3 orbital rotational symmetry in spin-singlet sys-tems, if the gap does not possess the full symmetry of thelattice. Furthermore, massive collective modes can ariseif the energy gap is degenerate or has an admixture of dif-ferent representations of the point group. The massivemodes can in principle lie below the gap edge, and canthus be relevant for the low frequency dynamics of cor-relation functions. In fact, Raman-active modes in spin-triplet superconductors such as Sr2RuO4 have drawn re-cent theoretical interest (Kee et al., 2003), although theexperimental challenges are not negligible because of thelow Tc and the related small energy gap in these materi-als.

Though very interesting, spin-triplet pairing or spin-orbit effects are rare and more on the exotic side insuperconductivity. Competing ground states, however,are quite common whenever correlation effects come intoplay. This is not at all confined to the cuprates, butoccurs also in, e.g., spin (SDW) and charge (CDW) den-sity wave systems. Usually, density-wave formation withlong-range order at least partially suppresses supercon-ductivity such as in 2H − NbSe2 (see below). Then, ad-ditional modes appear as a result of the competition be-

29

tween CDW ordering and superconductivity, and collec-tive modes appear as one modulates either one or bothorder parameters.

Littlewood and Varma (1981, 1982) and Browne andLevin (1983) considered a direct coupling between chargedensity and superconducting gap amplitudes, modulatedfor example by a CDW phonon, although this was notspecified. They obtained an additional “gap” mode be-low 2∆. Yet this mode was only considered in the L = 0channel, and Coulomb interactions once again removethis mode. Lei et al. (1985) considered an effective CDW-SC coupling via a phonon and realized that in anisotropicsystems such collective modes in L = 0 channels mayappear. Finally, Tutto and Zawadowski (1992) treatedelectron-phonon and CDW amplitude-phonon couplingon equal footing in finite angular momentum channels,showing generally that the collective modes in these chan-nels are unaffected by Coulomb screening. The modesobtained split off from the gap edge and appear as exci-tations below the quasiparticle spectrum, much like ex-citons in semiconductors. Evidence of mixed CDW-SCpairing may be seen in Raman experiments via the pres-ence or absence of these modes. This has recently beenextended by Zeyher (2003) to MgB2, having multiple en-ergy gaps on different electron bands.

The collective mode spectrum of coupled d−wavecharge density and superconductivity was investigatedby Zeyher and Greco (2002) along the lines developed byTutto and Zawadowski (1992) for conventional CDW andsuperconducting systems. As for s−wave CDW super-conductors, collective modes split off from the maximumof the gap edge. As an important difference in d−wavesystems, the modes distinctly affect the various symme-try channels. Besides the reorganization of A1g spec-tral weight, additional modes alter the B1g spectrum, asshown in Figure 23.

Density waves instabilities need not necessarily com-pete with superconductivity but rather can provide an ef-fective coupling mechanism (Castellani et al., 1995; Peraliet al., 1996) as long as quantum and thermal fluctuationssuppress long range order. Then, collective modes mayappear as fluctuation-induced modes. The Raman scat-tering is usually determined from Aslamazov-Larkin fluc-tuation diagrams considered for the conductivity (Asla-mazov and Larkin, 1968). To overcome the q = 0 phasespace limitations, the Raman response is given by theexchange of two fluctuations modes at wavevectors Qc

and −Qc, yielding generally a mode at energies of twicethe mass of the fluctuation propagator. Once again thepolarization dependence can select different fluctuationmodes corresponding to different ordering wavevectorscoupling to either charge or spin density modes. Thiswas investigated for spin (Brenig and Monien, 1992) andcharge (Caprara et al., 2005) fluctuations in the normalstate, and for novel spin resonances in the superconduct-ing state of the cuprates (Chubukov et al., 1999, 2006;Venturini et al., 2000).

Lastly, we remark that many other types of collective

modes are possible if one considers more exotic groundstates with different symmetry classifications. For ex-ample, a chiral spin liquid has been investigated byKhveshchenko and Wiegmann (1994) in which helical ex-citations were conjectured to exist and are in principlemeasurable in A2g orientations which can be projectedout via proper sums of spectra taken with both linearlyand circularly polarized light. Other examples are modesinduced by magnetic fields or optical modes resultingfrom Dzyaloshinskii-Moriya interactions in Heisenbergantiferromagnets as observed recently in lightly dopedLa2−xSrxCuO4, 0 ≤ x ≤ 0.03 (Gozar et al., 2005b) anddiscussed by Silva Neto and Benfatto (2005), directlydemonstrating the importance of spin coupling to thelocal environment.

III. FROM WEAKLY TO STRONGLY INTERACTINGELECTRONS

In this section we review experimental results in sys-tems other than doped semiconductors (see reviews by,e.g., Abstreiter et al. (1984) and Pinczuk and Abstreiter(1989)) and cuprates (see section IV) with a view to-wards signatures in the Raman spectra arising from thedevelopment of strong electronic correlations. We discussvarious types of superconductors and summarize resultson correlated metals and other strongly interacting sys-tems.

The light scattering cross section in absorbing media,such as systems with free carriers, is generally weak sincethe interaction volume is small for the short penetration

0.00 0.05 0.10 0.15 0.20frequency

0.0

0.1

0.2

0.3

0.0

0.1

0.2

0.3

−im

ag. p

art o

f χ(

ω)

0.0

0.2

0.4

0.6 δ=0.178

δ=0.114

δ=0.077

B1g

B1gB2g

B1g

B1gB2g

B1g

B1gB2g

(0)

(0)

(0)

FIG. 23 Electronic Raman spectra of B symmetries for threedifferent doping levels including coupling of d-CDW and d-superconducting amplitudes. The superscript 0 denotes thespectra in the absence of collective modes. From Zeyher andGreco (2002).

30

depth of visible light, δ λi = 2πc/ωi. As a conse-quence, the momentum perpendicular to the surface isnot conserved, and the transfer q is not given any moreby the difference of the vacuum momenta of the involvedphotons ki−ks but essentially by δ = λ/(4πk) with k theimaginary part of the index of refraction (Abrikosov andFal’kovskiı, 1961; Mills et al., 1970). Even in strongly ab-sorbing materials with k > 1, 1/δ π/a holds where ais the lattice constant, and the limit of small momentumtransfer is still effective. This introduces a new energyscale vF q ≈ vF /δ with vF and q being the magnitudesof the Fermi velocity and the momentum transfer, respec-tively. In all considerations, this scale must be put intorelation to the other relevant energies, such as the elec-tron scattering rate Γ = /τ in the normal and the gap ∆in the superconducting state. These apparent academicconsiderations have major impact on both the observ-ability and the interpretation of electronic spectra.

A. Elemental Metals and Semiconductors

In addition to the small scattering volume due to theabsorption of light by free carriers, a parabolic dispersionand a spherical Fermi surface reduce the cross sectionof single-electron excitations in metals and degeneratesemiconductors strongly, since in such systems the asso-ciated density fluctuations are screened by the long-rangeCoulomb interaction. The few spectra we are aware ofhave been taken on elements with a more complex bandstructure such as Nb (Klein, 1982a; Klein and Dierker,1984) or Dy (Klein et al., 1991). In Dy a broad con-tinuum similar to that in the high-Tc cuprates (Bozovicet al., 1987) is found. In Nb the superconducting statewas studied. Due to the low transition temperature Tc,the correspondingly small energy gap ∆(T ) and the smallratio δ/ξ with δ the penetration depth of the light andξ the superconducting coherence length (for details seesection III.B) the characteristic redistribution of scatter-ing intensity is very hard to observe. The peaks foundat 1.8 K in the expected energy range close to 2∆(T ) arevery weak, and no normal state spectra have been mea-sured for comparison (Klein, 1982a; Klein and Dierker,1984).

In fact, superconductors rather than normal metalswere the main focus in the early days of electronic Ra-man scattering. Only after the discovery of the cuprates(Bednorz and Muller, 1986), with generally complicatedand sometimes very surprising electronic properties, didstudies of the normal state become increasingly attrac-tive (see section IV).

B. Conventional Superconducting Compounds

Among superconductors, intermetallic compounds likeNb3Sn or V3Si with A15 structure, can be consideredconventional both above and below Tc. They are strictly

3D, superconductivity is mediated by phonons leading toan essentially isotropic s-wave gap, and correlations arebelieved to be of minor importance. This does not meanthey are simple. For instance, the Fermi velocity is verysmall, close to the velocity of sound, and the Fermi sur-face is multi-sheeted. Sufficiently perfect single crystalsof Nb3Sn and V3Si undergo a structural transformationfrom a cubic to a tetragonal lattice at low temperature.Nevertheless, A15 compounds are paradigms of strong-coupling s-wave superconductors with a high density ofelectronic states at EF . Materials like the borocarbides,MgB2 or 2H − NbSe2 are certainly more complex andcorrelations or multiband aspects come into play.

1. A15 compounds

Superconductivity induced structures close to twicethe gap edge were found in mono-crystalline Nb3Sn(Fig. 24) and V3Si (Dierker et al., 1983; Hackl et al.,1982, 1983; Klein, 1982a) two years after the discoveryof gap modes in 2H−NbSe2 by Sooryakumar and Klein(1980) (section III.C) and after an early but unsuccess-ful attempt in polycrystalline Nb3Sn by Fraas (1970).For T < Tc the scattering intensity is redistributed

FIG. 24 Raman spectra of Nb3Sn. Lower curves in (a) and(b) are at 40 K and upper curves are at 1.8 K. Data in (c)and (d) are at 1.8 K. Below Tc = 18 K the intensity at lowenergies is strongly suppressed with respect to the normalstate. Beyond a threshold of approximately 50 cm−1 a newpeak appears. The symmetries are Eg (a, d), T2g (b), Eg+A1g

(c top), Eg (c middle), and A1g (c bottom). The A1g datain (c) are obtained by subtracting the middle from the uppercurve. All solid lines except for the upper two in (c) aretheoretical fits to a broadened Maki-Tsuneto function (seesection II.D.6). From Dierker et al. (1983).

31

with a suppression below and a pile-up at approximately2∆ 50 cm−1. The well-defined peak in Eg symmetryfollows the BCS prediction for the temperature depen-dence of the gap up to approximately 0.85 Tc (Hacklet al., 1983, 1989). Somewhat unexpectedly, the peakfrequencies of the superconductivity-induced features de-pend on the selected symmetry (Table II). Independentof minor differences in the absolute numbers stemmingfrom the data analysis the Eg peaks are significantlybelow those having A1g and T2g symmetries. At firstglance one could think of a gap anisotropy to manifestitself. However, there is no support from the tunnelingresults which rather indicate the possible gap anisotropyto be opposite in V3Si and Nb3Sn and very large or fromcalorimetric studies which should track the smallest gap(Table II). In addition, the shapes of the Raman spectraare strongly symmetry dependent in that the Eg peakis much narrower than the others. The meaning of thisanisotropy was a matter of intense discussion.

The results in A1g scattering symmetry in Nb3Sn(Dierker et al., 1983; Klein, 1982a) and later in V3Si(Hackl and Kaiser, 1988a) demonstrate clearly that thestructures below Tc originate in light scattering fromCooper pairs (Fig. 24), since there exist no Raman ac-tive excitations at this symmetry, such as phonons orother bosonic modes in the A15 structure from whichthe electrons can borrow intensity. There is not even anelectronic continuum above Tc (see Fig. 24 (c) and Hackland Kaiser (1988a)). In spite of similar band structuresand densities of states at the Fermi level, N(EF), (Kleinet al., 1978) the intensities of the modes are quite dif-ferent in the two compounds as is the overall scatteringcross section. For this reason, the weak A1g mode in V3Siescaped detection for a while (Hackl and Kaiser, 1988a).Since there is nothing to interact with the peak frequen-cies, the A1g structures should be close to the energy gapin the respective material. One actually observes coin-cidence of both the A1g and T2g Raman energies withthose of bulk methods such as calorimetry and neutrons,while the Eg energies are substantially lower (Table II).

We first note that surface sensitive methods such astunneling return somewhat smaller gap energies thanbulk methods. Optical spectroscopy results are alsosmaller most likely due to surface treatment. Strainor disorder can indeed reduce Tc in A15 materials sinceN(EF ) decreases rapidly (Mattheiss and Weber, 1982).Similar reasons might apply for the Raman data in Nb3Snof Dierker et al. (1983) although the fits (see Figure 24)reveal gap values slightly below (5–10 %) the peak posi-tions. Spectra of cleaved surfaces, such as those of V3Siand Nb3Sn taken by Hackl and Kaiser (1988a) and Hacklet al. (1989), respectively, apparently give gaps closer tothe bulk values.

For these reasons, it seems worthwhile to look for othersources of the anisotropy, and we consider an interpreta-tion in terms of final state interactions (Bardasis andSchrieffer, 1961; Klein and Dierker, 1984; Zawadowski etal., 1972). This means that the two single electrons of

Sample Raman Energy (cm−1) Reference Data (cm−1)

A1g Eg T2g

50 tunneling e

35–13 tunneling f

Nb3Sn52 41 50 a

53 tunneling g67 48 70 b

62 calorimetric h

56 neutrons i

37 tunneling j

40–50 tunneling k

V3Si- 40 - c

46 IR l55 42 52 d

41 IR m

49 calorimetric h

TABLE II Gap energies in A15 compounds as measured byRaman scattering and other methods. a and c refer to re-sults from fits (see Fig. 24 and section II.D.6), b and d arepeak frequencies. In two cases an anisotropy was found bytunneling being indicated by a range (f and k). The firstand the second numbers are for [100] and [111] directions,respectively. Results of the following publications are used:a (Dierker et al., 1983), b (Hackl et al., 1989), c (Klein andDierker, 1984), d (Hackl and Kaiser, 1988a), e (Rudman andBeasley, 1984), f (Hoffstein and Cohen, 1969), g (Geerk etal., 1984), h (Junod et al., 1983), i (Axe and Shirane, 1973),j (Moore et al., 1979), k (Morita et al., 1984), l (Tanner andSievers, 1973), m (Perkovitz et al., 1976).

a broken Cooper pair can still interact in channels or-thogonal to the pairing channel. The strongly coupledEg phonon (Schicktanz et al., 1980, 1982; Weber, 1984;Wipf et al., 1978) is in fact orthogonal to the fully sym-metric (s-wave) pairing channel. Hence, it is capable offorming a bound state below the pair-breaking threshold,

FIG. 25 Raman spectra in Eg symmetry of V3Si. FromMonien and Zawadowski (1990).

32

explaining both the reduced energy and the linewidth ofthe Eg gap mode (Monien and Zawadowski, 1990). Fitsto the results in V3Si are substantially improved by in-cluding the bound state (Fig. 25) in comparison to thoseneglecting it (Klein and Dierker, 1984). Additional ex-perimental support comes from the evolution with tem-perature of the spectra in V3Si and Nb3Sn (Hackl et al.,1983, 1989). In either compound, the integrated spectralweight in A1g symmetry increases significantly because anew scattering channel opens up below Tc due to the for-mation of Cooper pairs while staying essentially constantin Eg symmetry because the weight is being transferredfrom the phonon to the bound state (Fig. 26).

In contrast to Eg symmetry, the pair-breaking featuresin T2g symmetry are weak and essentially at the A1g po-sition. The question arises as to why there is no boundstate although there exists a phonon. Clearly, the T2g

phonon intensity is weak and the line width is small,reflecting the moderate coupling as opposed to Eg sym-metry where the complete line width and the asymmetricFano shape stem from the coupling to conduction elec-trons (Weber, 1984; Wipf et al., 1978). The bound state’senergy split-off by approximately 30 % indicates that thevery strong interaction drives the system close to an in-stability of the s-wave ground state. On the other hand,the T2g mode is only weakly coupled and the interactionwith the conduction electrons is not strong enough tosubstantially renormalize the spectrum.

Symmetry arguments, the unique line shape and theintensity transfer in Eg symmetry, as well as the com-parison to calorimetric results, make us believe that theformation of a bound state is more likely an interpreta-tion of the Eg results in A15 compounds than the man-

FIG. 26 Raman spectra of Nb3Sn at Eg (a) and A1g (b) sym-metries. The integrated spectral weight (limits indicated byarrows) stays constant to within 3 % in Eg symmetry whileincreasing by a factor of 3 in A1g symmetry on cooling fromTc to 6 K. For clarity the data points are omitted and onlythe results of a smoothing procedure are displayed. The scat-ter of the data is smaller than 0.1 units around the lines.From Hackl et al. (1989), reproduced with permission fromElsevier, c© 1989.

ifestation of a gap anisotropy or a two-gap scenario. Incontrast, both effects may cooperate in 2D MgB2 discov-ered to be a superconductor just recently by Nagamatsuet al. (2001).

2. MgB2 and the Borocarbides

Electronic Raman studies on MgB2 have explored thesuperconducting energy gap and changes in phonon line-shapes occurring below Tc, starting with the work ofChen et al. (2001) and followed thereafter by Quilty et al.(2002, 2003). Here the symmetry dependence of the re-sponse allowed a direct observation of the pairing gapon the two-dimensional σ bands and the 3D π bands. Byorienting the light polarizations along the c-axis of MgB2

(perpendicular to the hexagonal planes) the σ bands can-not be probed for having little dispersion and thus the πbands are projected out, giving a value 2∆π = 29 cm−1.Other polarizations are able to detect a larger pairinggap 2∆σ = 100 cm−1. zz-polarized spectra in the super-conducting state are shown in Figure 27 along with fitsfrom the theory for disordered s−wave superconductors.

The gap values are consistent with those from othertechniques, yet the fit yields values of the disorder-relatedscattering rate different from those of the resistivity bya factor of 2 (Quilty et al., 2003). Zeyher (2003) has re-analyzed the fit where the direct coupling of light to theσ band is zero and the σ gap appears as a result of acoupling to the Raman active E2g phonon, believed tobe largely responsible for pairing. The xx spectrum inthe superconducting state can be understood then as asuperposition of a phonon line, a background, and a col-lective bound state due to residual interactions betweenelectrons, similar to that observed in A15 compounds (seeFigures 25 and 26 (a)).

The superconducting energy gap has also been stud-

2.0

1.5

1.0

0.5

0.0

Inte

nsity

(ar

bitr

ary

untis

)

4003002001000

Raman Shift (cm-1)

0.3

0.2

0.1

0.0Inte

nsity

(ar

b. u

ntis

)

4003002001000Raman Shift (cm-1)

zz 15 Kzz Fit

xx 15 Kxx Fit

FIG. 27 Raman spectra for xx and zz polarization geometriesin the superconducting state of MgB2 taken by Quilty et al.(2003). The solid lines are fits to the data using the theoryof Devereaux (1992) for disordered s−wave superconductorsusing a single gap and two gap model for zz and xx polariza-tions. The xx spectrum has also been interpreted in terms ofa collective bound state by Zeyher (2003).

33

ied in some detail in the borocarbide superconductorsRNi2B2C (R=Y, Lu) by Yang et al. (2000a,b). Sharp 2∆peaks were observed in A1g and B2g symmetries, whilethe maximum in B1g symmetry is less pronounced and20% higher in energy. All peaks showed a typical BCS-type temperature dependence and disappeared above theupper critical field Hc2. Due to the high surface qualityand improved instrumentation, the residual scattering in-tensity below the gap edge is much smaller than, e.g., inthe A15 compounds but finite with an approximately lin-ear variation with energy.

Since a direct coupling to a Raman active mode wasnot found in the borocarbides, it is more complicatedthan in the A15 compounds or in MgB2 to sort outwhether a gap anisotropy, multi-gap superconductivityor collective modes are responsible for the variations inenergy and line shape at the different symmetries. Ofcourse, the existence of bound states is not related toRaman-active modes; only the experimental verificationis more indirect, e.g., via the line shape (see Figure 25).Similarly, the linear low-energy scattering can suggesteither the presence of strong inelastic scattering due tolarge coupling constants λ (Allen and Rainer, 1991; Yanget al., 2000b) or gap nodes in pairing states with lowersymmetry, such as s+g-wave superconductivity (Lee andChoi, 2002). A quantitative analysis on the basis of a re-alistic band structure could possibly help clarify these,at present, open issues.

While superconductivity was the dominant correlationin the A15 compounds, MgB2 was more complex due tothe interplay between 2D and 3D behavior. In the boro-carbides magnetic order as a second instability compet-ing with superconductivity comes into play (Canfield etal., 1998). Although the Raman studies were performedon non-magnetic compounds (Yang et al., 2000a,b) thevicinity of different types of order is characteristic for thisand the following classes of systems.

C. Charge Density Wave Systems

The competition or coexistence of different groundstates was studied intensively in layered dichalcogenidesin the 1970s and 1980s. The interest in these chargedensity wave (CDW) systems was revived after the dis-covery of superconductivity in the cuprates for two rea-sons: in both compound classes superconductivity com-petes with one or more other instabilities, and, secondly,the dramatically improved instrumentation allowed qual-itatively new and unexpected insights into materials like2H−NbSe2. As an example, the electronic scatteringrate τ−1 exhibits marginal (Varma et al., 1989a) ratherthan Fermi liquid like temperature and energy depen-dences (for a discussion and for references see, e.g., Cas-tro Neto (2001)).

1. 2H−NbSe2

2H−NbSe2 is a layered, though 3D, superconductorwith an in-plane coherence length ξ‖ of approximately70 A and ξ⊥ = 25 A (de Trey et al., 1973). The pene-tration depth for visible light δ is of the order of 200 A,hence ξ⊥ δ. The discontinuity at 2∆ is expected to in-crease with 2∆/(vF q) ≈ δ/ξ (Klein and Dierker, 1984).In addition, the material can be cleaved easily, facilitat-ing the preparation of atomically flat surfaces from whichdiffuse scattering of laser light is minimized. These arefavorable (though not easy!) conditions for observing gapstructures close to the elastic line.

In Fig. 28, the first observation of the redistributionof scattering intensity in the superconducting state of2H−NbSe2 with the sample immersed in superfluid He isreproduced. The effect is measured for two samples withslightly different impurity concentrations. In either case,the fully symmetric A and the E responses are shown.13The peaks have slightly different energies, and are locatedat 18 and 15 cm−1, respectively, close to the essentiallyk independent leading edge gaps found in recent photoe-mission experiments (Valla et al., 2004). In the normalstate at 9 K the new low energy modes are absent, whilethe maximum related to the CDW state is still present.

FIG. 28 Raman spectra of 2H−NbSe2 above and below Tc.The lower curves of each panel are taken in the normal state(9 K) the upper ones at 2 K well below Tc with the sampleimmersed in superfluid He. At the A (‖ − ⊥) and the E (⊥)symmetry the superconducting spectra are offset by 40 andby 20 counts, respectively. According to the strength of theCDW mode (labeled by C) Sample B and M have slightlydifferent impurity concentrations. From Sooryakumar andKlein (1980).

13 In the (incommensurate) CDW phase the symmetry representa-tions A1g etc. of the D6h point group do not apply any more.

34

FIG. 29 Raman spectra in A symmetry of 2H−NbSe2 (sam-ple B) in various magnetic fields at 2 K. From Sooryakumarand Klein (1980).

As can be seen in all panels, the CDW mode hardensbelow the superconducting transition.

The difference between the two samples is apparentlythe impurity concentration affecting the strength of boththe CDW and the gap mode. In fact, the CDW transi-tion can be suppressed by either pressure or an increas-ing number of defects, which may be quantified by theresidual resistance ratio (Huntley and Frindt, 1974). In asystematic study of impurity effects, Sooryakumar et al.(1981) showed that the gap excitations go away alongwith the CDW mode while the superconducting transi-tion temperature is essentially unchanged. It is temptingto assume that the gap modes are directly coupled tothe CDW mode and exist only along with it. This inter-pretation is supported by results obtained in a magneticfield (Fig. 29). Upon increasing the field the gap fea-ture in A symmetry is gradually suppressed while theCDW mode gains intensity leaving the energy integralover the Raman response χ′′(ω) constant to within 7%(Sooryakumar and Klein, 1981). In E symmetry no clearsum rule could be found (Sooryakumar and Klein, 1981),and it is possible that some of the gap intensity appearsindependent of the CDW.

Particularly the result in A symmetry (Fig. 29), wherethe gap mode gains intensity at the expense of the CDWmode, triggered the theoretical work to follow.14 Theavailable data are not supportive of a sum rule in E sym-

14 For convenience we again give the related references being dis-cussed thoroughly in the context of collective modes and in theprevious paragraph: (Balseiro and Falicov, 1980; Browne andLevin, 1983; Klein and Dierker, 1984; Lei et al., 1985; Little-wood and Varma, 1981, 1982; Monien and Zawadowski, 1990;Tutto and Zawadowski, 1992). We would like to draw the read-ers’s attention also to the closely related Raman experiments insuperfluid 4He (Greytak and Yan, 1969) and their theoreticaldescription (Zawadowski et al., 1972).

metry (see Fig. 28 a), demonstrating similarities with theA15 compounds where also both electronic scattering andcoupling to phonons was observed. In order to bring somelight into the rather involved discussion, it is worthwhileto reconsider the influence of impurities (Sooryakumaret al., 1981).

At first glance, the reaction to disorder points in thesame direction as the results in magnetic fields. How-ever, defects not only suppress the formation of the CDW(Huntley and Frindt, 1974) but also, independently, re-duce the intensity close to 2∆ (Devereaux, 1992, 1993)while normally leaving the transition temperature Tc of aconventional s-wave superconductor unchanged (Ander-son, 1959)15. For vF /δ ∆, /τ the intensity at 2∆is proportional to τ∆ (Devereaux, 1992, 1993). In thisrespect Raman is just opposite to the optical conductiv-ity where the gap can be observed only if /τ is of theorder of ∆ or larger (Mattis and Bardeen, 1958). Thisimplies that the Raman gap feature can be wiped out byimpurities while Tc remains essentially constant; at thesame time, though independently, the CDW transition issuppressed. Hence, it is possible that the gap features in2H−NbSe2 exist on their own as pair-breaking effect butthe interaction with the CDW leads to a bound state.

Most of the other CDW systems are not supercon-ducting, but show very interesting behavior around thetransition to the charge-ordered phase. Some of themhave been studied earlier using light scattering. Here webriefly discuss a recent study of the temperature-pressurephase diagram of the CDW state.

2. 1T -TiSe2

In 1T -TiSe2 a commensurate CDW is established be-low TCDW 200 K. The amplitude of the CDW couplesto zone-boundary acoustic phonons which are folded tothe center below TCDW (Snow et al., 2003). Pronouncedsoft-mode behavior can be observed as a function of tem-perature. In the limit T → 0 two strong lines at 115 and75 cm−1 in A1g and Eg symmetry, respectively, dom-inate the low-energy spectra. By increasing the pres-sure the CDW state first stiffens along with the latticethen disappears rapidly in the pressure range of 5 to25 kbar. Above 25 kbar a quantum disordered (essen-tially isotropic) metallic or semi-metallic state is foundalthough the Raman continuum typical for a metal isnot reported. The quantum mechanical melting of theCDW order is in many ways similar to classical 2D melt-ing, with the appearance of crystalline and disorderedCDW regimes, as well as an intermediate “soft” CDW

15 In A15 compounds the high density of electronic states at EF

(partially responsible for the high Tc) depends sensitively on dis-order (Mattheiss and Weber, 1982). Hence, disorder reduces Tc

fast as opposed to what one would expect from the Andersontheorem.

35

regime in which the CDW exhibits strong fluctuationsand loses stiffness. Here, measurements on the develop-ment and polarization dependence of the electronic con-tinuum raises the possibility of following quantum criticalbehavior in other systems with competing orders. Thisis a promising direction for future studies.

D. Kondo or Mixed-Valent Insulators

All correlations discussed so far are related to electron-lattice interactions in systems with screening lengths ofthe order of the interatomic spacing. With decreasingelectronic density, new phenomena develop originatingfrom the competition between kinetic and potential en-ergy of the conduction electrons. Well known examplesare the Mott metal-insulator transition or Wigner crys-tallization. In either case, the material becomes an in-sulator at low temperature due to “immobilization” ofelectrons rather than an energy gap as in band insula-tors.

Raman scattering on band insulators and semiconduc-tors has been well-documented, with a focus placed onhigh energy charge transfer excitations. Yet the devel-opment of the Raman response at low frequencies canin principle shed light on the development of electroniccorrelations with temperature and/or doping.

Experiments by Nyhus et al. (1995a) on Kondo insulat-ing FeSi shown in Fig. 30 and Nyhus et al. (1995b, 1997a)on mixed-valent SmB6 have indeed shown the transfer ofspectral weight from low to high energies as the temper-ature is lowered into the insulating state. This “univer-sality” suggests that there is a common mechanism gov-erning the electronic transport in correlated insulators.As these materials are cooled, they all show a pileup ofspectral weight for moderate photon energy losses witha simultaneous reduction of the low-frequency spectralweight. This spectral weight transfer is slow at high tem-peratures and then rapidly increases as the temperatureis lowered towards a putative quantum-critical point cor-responding to a metal-insulator transition.

A characteristic energy appears which separates theregion of intensity depletion from intensity enhancementwith temperature. This characteristic frequency is es-sentially independent of temperature and is thus calledan isosbestic point in the spectra (as shown in Figure30). Finally, it is often observed that the range of fre-quency where the Raman response is reduced as T islowered is an order of magnitude or more larger than thetemperature at which the low-frequency spectral weightdisappears. As discussed in the Section II.D.4 these find-ings are consistent with the loss of low-frequency scatter-ing due to the thermal depletion of excited states. Thechannel-dependence has not yet been a focus of interest.If it were measured, important information concerningthe evolution of the potential energy with doping couldbe inferred (Freericks et al., 2005).

FIG. 30 Temperature dependence of the Raman responsemeasured in FeSi by Nyhus et al. (1995a). Here ∆c denotethe position of energy gap developing in the continuum at lowtemperatures, transferring spectral intensity into the peak atenergy ∆0 . The sharp low energy features are phonons.

E. Magnetic, charge and orbital ordering: Ramanscattering in Eu-based compounds, Ruthenates, and theManganites.

There has been a great deal of interest in the relation-ship between diverse and exotic low-temperature phasesof strongly correlated systems (Dagotto, 2005). In par-ticular, the manganites, ruthenates, Eu-oxides, and hexa-borides display charge ordered, paramagnetic insulating,and ferromagnetic metallic phases as a function of dop-ing, temperature, and/or temperature. Due to the com-plex interplay between spin, charge, and orbital degreesof freedom, these systems present a battleground wheredifferent ordered phases compete for primacy as knobs ofthe Hamiltonian are changed (Imada et al., 1998).

The phase diagram of strongly correlated materials ismore complex in systems having strong electron-latticeinteractions as well as orbital ordering tendencies. Ra-man spectroscopy in systems such as the manganites andruthenates have provided important information on theevolution of lattice, charge, and spin dynamics acrossphase boundaries. In many cases the transitions can beinduced by applying pressure.

While Raman scattering from phonons has tradition-ally provided important information concerning the de-velopment of locally or globally symmetry-broken statesaccompanied by the formation of static charge ordering,electronic Raman spectroscopy can be brought to bearon this problem as it can detect both fluctuating or sta-tic charge and/or spin ordering, and may reflect on thetendency toward orbital ordering as well. In addition,the polarization dependence can shed light on the typesof excitations that are created in or near the orderedphases which may serve as signatures that certain inter-actions are more prominent than others. Thus Raman is

36

a powerful spectroscopic method by which the dynamicsacross quantum phase transitions can be investigated incorrelated systems.

Recent Raman studies on EuB6 (Nyhus et al., 1997b),Eu1−xLaxB6, EuO (Snow et al., 2001), and Eu1−xGdxO(Rho et al., 2002), show that the metal-semiconductortransition in these materials is accompanied by distinctchanges of the electronic continuum. A high-temperatureparamagnetic semimetallic phase is well characterized byscattering from diffusive charge excitations which becomeless diffusive at lower temperatures when correlation ef-fects have not yet set in. However, the diffusive scatter-ing rate, when fit with Eq. (49), increases with decreasingtemperature and scales with the magnetic susceptibilityas the system begins to develop short-range magnetic or-der, typical of insulating behavior, as shown in Figures 19and 30. Finally, at low temperatures, a ferromagneticmetallic phase occurs, showing a flat continuum charac-teristic of a strongly correlated metal. The doping, po-larization, and magnetic field dependence of the spectraimplies that the metal-semiconductor transition is pre-cipitated by the formation of bound magnetic polaronsabove the ferromagnetic ordering temperature.

Concerning the ruthenates, recently Snow et al. (2002)and Rho et al. (2003) have studied the evolution of thespin and lattice dynamics through the pressure-tunedcollapse of the antiferromagnetic Mott-like phases ofCa2RuO4, Ca3Ru2O7, and Ca2−xSrxRuO4 into a ferro-magnetic and possibly orbitally ordered metallic state atlow temperatures. The studies have shown many char-acteristic features resulting from the interplay of strongelectron-lattice and electron-electron interactions. Theseinclude (i) evidence of an increase of the electron-phononinteraction strength, (ii) an increased temperature de-pendence of the two-magnon energy and linewidth inthe antiferromagnetic insulating phase, (iii) evidence ofa charge gap development significantly below the metal-insulator transition (TMI), and (iv) a hysteresis asso-ciated with the structural phase change. The latter twoeffects are indicative of a first-order metal-insulator tran-sition and a coexistence of metallic and insulating compo-nents for T < TMI. The measurements have not yet beenextended to probe the unconventional superconductingstate at low temperatures.

Raman measurements on cubic and layered manganiteshave been used to explore the interplay of spin, chargeand orbital degrees of freedom. Yamamoto et al. (2000)and Romero et al. (2001) observe a suppression of thelow-energy continuum at B1g symmetry upon enteringthe charge- and orbital-ordered state. The interpreta-tion is not yet settled. Possible candidates are spin den-sity or dynamical charge-orbital fluctuations (Yamamotoet al., 2000) or a collective CDW excitation Romero et al.(2001). The controversy can probably not be solved with-out a quantitative theoretical description.

Bjornsson et al. (2000) have measured cubicLa1−xSrxMnO3 and via phonon lineshape analysis haveshown strong electron-phonon interactions involving lo-

cal lattice distortions in the high temperature paramag-netic state which gradually vanish below the ferromag-netic transition. A broad hump in the electronic spectradevelops at low temperatures in the metallic state around400 cm−1 for x=0.2, and weakens in intensity and shiftsto higher energies for x=0.5. Although a polaronic peakwould weaken with increasing carrier density, the shift to-wards higher energies with doping was interpreted ratheras a low-energy plasma excitations within the ferromag-netic metallic phase. Since recent ARPES studies onthe same compound (Mannella et al., 2004) and bi-layermanganite (Mannella et al., 2005) have revealed coexis-tence of quasiparticle and polaron features in the metallicphase and do not show evidence for low energy plasmaexcitations, more work is needed to clarify this issue.

A comprehensive study via reflectance and Ra-man measurements on Pr0.7Pb0.21Ca0.09MnO3,La0.64Pb0.36MnO3, La0.66Pb0.23Ca0.11MnO3, andPr0.63Sr0.37MnO3, by Yoon et al. (1998), have shownthat the electronic continuum displays a change fromdiffusive polaronic peaks at high temperature to aflat featureless continuum, similar to that observedin the cuprates, in the low temperature ferromagneticphase. A broad polaronic peak around 1200 cm−1 shiftsto lower energies with increases doping indicative ofweakened polaron binding energies. This is consistentwith a crossover from a small-polaron-dominated regimeat high temperatures to a large-polaron-dominatedlow-temperature regime. The low temperature phasealso provided evidence for the coexistence of large andsmall polarons, also in agreement with the ARPESresults.

Recent work on Bi1−xCaxMnO3 (x < 0.5) by Yoonet al. (2000) is of particular interest in connection withpolarization studies for the evidence of charge ordering.As shown in Figure 31, Raman scattering offers a uniquemeans of probing the unconventional spin and/or chargedynamics that arise when charge carriers are placed inthe complex spin environment of a charge-ordered sys-tems. Using circularly (LL) in addition to linearly (xxand yy) polarized light anti-symmetric components ofthe Raman tensor were isolated. In cubic crystals suchas Bi1−xCaxMnO3 they transform as the T1g irreduciblerepresentation (equivalent to A2g in tetragonal materi-als like the cuprates). Upon entering the charge-orderedphase a quasielastic scattering response appears with theT1g symmetry of the spin-chirality operator. Thus it wasconjectured that the chiral excitations were signaturesof either a chiral spin-liquid state associated with theMn core spins, or of closed-loop charge motion causedby the constraining environment of the complex orbitaland Neel textures. A possible path for charge motion isshown in Figure 31, emphasizing the circular nature ofcharge transfer. It is remarkable, that the spectral shapeand the temperature variation of the characteristic en-ergy are quite similar to the low-energy response in thecuprates (see section IV.D.3) although a state with sta-tic order is entered in Bi1−xCaxMnO3. It is interesting

37

and perhaps important how the two types of response arerelated.

Recently Saitoh et al. (2001) have performed Ra-man measurements on detwinned and orbitally orderedLaMnO3 and have observed multiple peak structureswhich they interpret as orbital excitations or “orbitons”.While this has been challenged by Gruninger et al. (2002)on the basis of selection rules, more recent measurementsby Kruger et al. (2004) related the peak features to sec-ond order phonon scattering activated via the Franck-Condon mechanism (Perebeinos and Allen, 2001). Even

40801201600100200300400500Im00.10.20.30.40.50.60.700.511.52T / TCO010020030000.20.40.60.8100.20.40.60.811.2

39

Slakey et al., 1990b) visualized strong anisotropic inter-actions in these materials. Specifically, the observationof a polarization-dependent gap opening in the spectrabelow Tc was instrumental in solidifying the symmetryand orientation of the superconducting order parameter.Finally, more recent work has focused on elucidating thebehavior of competing orders and complexity which comehand-in-hand with strong correlations.

It is safe to say that no other system has been studiedso intensely via a bevy of experimental tools in recentyears as the cuprates. This is certainly true for Ramanspectroscopy, where the light scattering cross sectionshave been analyzed in a rich number and quality of ma-terials. In this section, we discuss Raman results on thecuprates and related materials, with an overview of re-lating findings to the physics uncovered in other systems.We show issues in which consensus has been reached andother issues which are controversial and require furtheranalysis.

A. From a Doped Mott Insulator to a Fermi Liquid

The so-called parent compounds of the cuprate super-conductors are antiferromagnetic Mott insulators. Athalf filling, charges are completely localized and excitedstates are separated by the Coulomb energy U prohibit-ing double occupancy in the Hubbard model.18 Theground state is a Heisenberg antiferromagnet which de-velops 3D long-range order below a Neel temperature oftypically 300 K. The excitations are spin waves with avery small (in-plane) anisotropy gap at q = 0 and anearest-neighbor exchange coupling J . The spin waveshave been studied right at the beginning by Raman scat-tering, and J was determined from the two-magnon peakto be of order 100 meV.19 Very early the existence ofchiral spin excitations, i.e. excitations where the spinsare rotated out of the easy (CuO2) plane, was demon-strated (Sulewski et al., 1991). The resonance profile20

has been described in terms of the triple resonance theoryof Chubukov and Frenkel (1995a,b). Very recently, onemagnon excitations at q = 0 were observed in the Neelstate of La2−xSrxCuO4 (Gozar et al., 2004) and relatedto Dzyaloshinskii-Moriya and XY optical modes result-ing from a spin gap by Silva Neto and Benfatto (2005).The experiments are the q = 0 and Ω → 0 manifesta-tion of canted spins which give also rise to scattering inthe two-spin-flip channel at large energies in A2g symme-try (Sulewski et al., 1991). Here, a lot more information,

18 The subject has been reviewed in detailed articles and books(Fulde, 1995; Gebhard, 1997)

19 (Blumberg et al., 1996; Knoll et al., 1990; Lyons et al., 1988,1989; Rubhausen et al., 1997; Sugai et al., 1988; Sulewski et al.,1990, 1991)

20 A selection of references is (Blumberg et al., 1996; Knoll et al.,1996; Lyons et al., 1988; Rubhausen et al., 1997).

such as the Dzyaloshinskii-Moriya vector, the size of theanisotropy gap, and the spin-lattice interaction strength,could be derived (Gozar et al., 2005b).

If the insulator is doped off half filling, antiferromag-netic long range order disappears quickly with hole dop-ing but survives up to higher electron doping levels (Fig-ure 32). Yet short ranged antiferromagnetic correlationsare not quenched even at high doing levels, as seen by thepersistence of the two-magnon peak in B1g Raman scat-tering experiments on both the hole-doped (Blumberget al., 1994; Reznik et al., 1993; Rubhausen et al., 1999;Sugai et al., 2003) and the electron-doped side (Onoseet al., 2004) of the phase diagram. This is summarizedin Figure 33 for a number of compounds and shows howthe two-magnon peak softens and broadens with doping,eventually merging into the continuum at higher dop-ing levels. Since the two-magnon intensity is dominatedby a double spin flip of nearest neighbors, it can be ob-served even for small magnetic correlation lengths ξm ofthe order of few lattice constants (see Figure 17). Thetwo-magnon peak persists at least up to optimal dop-ing for hole-doped systems (Fig. 33). The position ofthe maximum and the peak intensity decrease by fac-tors of roughly 2 and 20, respectively, for 0 ≤ p ≤ 0.16.Consistent with the Raman results, neutron scatteringexperiments in La2−xSrxCuO4 revealed magnetic excita-tions in the complete superconducting range (Wakimotoet al., 2004).

The broadening of the two-magnon response with dop-ing and temperature has been studied theoretically onclusters for the t− J model (Prelovsek and Jaklic, 1998)and by quantum Monte Carlo (QMC) techniques (Sand-vik et al., 1998), respectively. Knoll et al. (1990) analyzedtheir experiments at elevated temperatures in terms ofspin-lattice coupling. While many features of the dataat 1/2 filling for undoped cuprates can be captured bystudies of HELF Eq. (46), in general the evolution ofthe magnon lineshape with doping and temperature andits full polarization dependence is not very well under-stood. In particular, it is an unsettled issue, how, whena more metallic state develops for higher doping levels,the magnon line merges into the relatively featurelesscontinuum (shown in Figures 33 and 34) that extendswell-beyond all relevant energy scales such as qvF, thesuperconducting energy gap ∆ or the maximal phononenergy ωD. In strongly overdoped yet superconduct-ing samples (0.20 < p < 0.27), the physical propertiesin the normal state are still not those of a conventionalmetal. The continuum itself displays significant polariza-tion and doping dependence, which will be addressed insection IV.D.

For p > 0.05 (n > 0.12), superconductivity emergesand reaches a maximal Tc at approximately p = 0.16 (n =0.14). In the Raman spectra, superconductivity inducedpeaks emerge out of the flat continuum in the normalstate, accompanied by spectral weight reorganization fortemperatures below Tc, as shown for Bi2Sr2CaCu2O8+δ

in Figure 34.

40

FIG. 33 Doping dependence of the two-magnon B1g spectra in a number of compounds at 300 K. From Sugai et al. (2003).

In the following section we focus on the polarizationdependence of the Raman response below Tc in bothhole- and electron-doped cuprates. We summarize howsymmetry arguments can be used to obtain the momen-tum dependence of electronic properties in general and,specifically, that of the energy gap ∆(k). The dopingdependence in hole-doped systems is postponed and willbe discussed in detail in section IV.C.

FIG. 34 Raman spectra in the normal state (a) and supercon-ducting state (b) for A1g + B2g (left panel) and B1g (+A2g)(right panel) orientations in Bi2Sr2CaCu2O8+δ. From Ya-manaka et al. (1988).

B. Superconducting Energy Gap and Symmetry

Early Raman results for the electronic continuumshowed only little difference between the normal and thesuperconducting states (Lyons et al., 1987). We knownow that the relatively high defect concentration in thefirst samples suppressed the structures related to thegap. The synthesis of flux-grown YBa2Cu3O7 with asufficiently small number of defects (Kaiser et al., 1987;Schneemeyer et al., 1987) improved the situation rapidly,and a clear indication of the pair-breaking effect was ob-tained by Cooper et al. (1988a) on YBa2Cu3O7, showingthe emergence of a peak and reorganized spectral weightoccurring for temperatures below Tc as expected fromtheory. Soon thereafter a strong polarization dependenceof the Raman spectra was observed (Fig. 34), which canbe considered the first spectroscopic evidence of a gapanisotropy (Cooper et al., 1988b; Hackl et al., 1988b;Slakey et al., 1990a; Yamanaka et al., 1988). In contrastto conventional materials (see, e.g., Fig. 24), there is nosharp onset of the scattering intensity at a threshold. Asan explanation of the continuous increase of the scatter-ing intensity at small frequencies the possibility of nodeswas discussed early (Hackl et al., 1988b; Monien and Za-wadowski, 1989). Originally shown in Bi2Sr2CaCu2O8+δ

and YBa2Cu3O7, the studies were quickly extended toother optimally hole doped cuprates. It was shown byKang et al. (1996, 1997) that magnetic fields suppressthe peaks, independently indicating their relationship tosuperconductivity.

41

1. Symmetry: B1g and B2g

The polarization dependence of both the peak fre-quencies and the low-energy slopes of the superconduct-ing spectra had been a vexing problem for several yearsfollowing the first observation of the reorganized spec-tral weight and the polarization-dependent response. Asshown in Figure 34, the peak in the B1g response devel-oped at roughly 30 percent higher energies than for B2g

or A1g polarization geometries, and the low frequencyspectra rose as Ω3 for B1g and linearly with Ω for otherchannels. In addition, the temperature dependent deple-tion at low frequency shifts was faster in B1g symmetrythan in other channels.

The satisfactory description emerged as soon as themomentum dependences of the Raman vertices of differ-ent symmetries µ, γµ(k), and of the energy gap ∆(k)were properly taken into account, as outlined in SectionII.D.6. As indicated by a few experiments before (Hardyet al., 1993; Shen et al., 1993) and corroborated by manymore later (Scalapino, 1995), a gap with dx2−y2 symme-try is the most compatible with the results in the cuprates(Devereaux et al., 1994a).

The predictions of d−wave theory - such as fre-quency power-laws, temperature dependences, and rel-ative peak positions - were found to be consistent

FIG. 35 Fit to the B1g and B2g data on as-grown (Tc=86 K,top panel) and oxygen-annealed (Tc=79 K, bottom panel)Bi2Sr2CaCu2O8+δ from Staufer et al. (1992). The inset isa log-log plot of the low frequency B1g response, showing across-over from linear to cubic behavior at a characteristic fre-quency ω∗/∆0=0.38 and 0.45 for the top and bottom panels,respectively. From Devereaux (1995b).

with many optimally hole-doped compounds, such asYBa2Cu3O7 (Chen et al., 1994a; Devereaux and Einzel,1995a), Bi2Sr2CaCu2O8+δ (Blumberg et al., 1997a; De-vereaux et al., 1994a; Devereaux and Einzel, 1995a),La2−xSrxCuO4 (Chen et al., 1994b), Tl2Ba2CuO6

(Blumberg et al., 1997b; Devereaux and Einzel, 1995a;Kang et al., 1996, 1997), Bi2Sr2CuO6+δ (Einzel andHackl, 1996), Tl2Ba2CaCu2O8 (Kang et al., 1997; Mak-simov et al., 1990), Tl2Ba2Ca2Cu3O10 (Hoffmann et al.,1994), and later HgBa2Ca2Cu3O8+δ (Sacuto et al., 1998,2000), Bi2Sr2Ca2Cu3O10+δ (Limonov et al., 2002a) and

42

Additional attributes have been intensely investigated,such as the agreement of the effective mass approxima-tion calculated within LDA for d−wave superconductorsin YBa2Cu3O7 (Strohm and Cardona, 1997), effects oforthorhombicity and mixing of s−wave components al-lowed by symmetry (Nemetschek et al., 1998; Strohmand Cardona, 1997), as well as the issue of the A1g peakseen in the superconducting state (Krantz and Cardona,1994), which is among the complicated though eventuallycrucial problems on the way to a better understanding ofthe cuprates.

2. The A1g problem – Zn, Ni and Pressure

While the B1g and the B2g spectra are in reasonableagreement with the predictions on the basis of d-wavepairing various problems arose in A1g symmetry whichslowed down the acceptance of the model in the begin-ning. One of the objections was the strong intensityfound in the A1g channel. As discussed in Section II.D.6,the A1g response is complicated due to the backflowterms which are as singular as the bare terms for frequen-cies at the gap edge, leading to cancellations of divergingintensities. For both a cylindrical Fermi surface and atight-binding band structure (Devereaux et al., 1994b;Krantz and Cardona, 1994), the effective mass approxi-mation predicts a suppression of the A1g intensity in com-parison to other channels, in contrast to what is foundexperimentally, as shown in Figures 34 and 38. Since theeffective mass approximation is of questionable utility inhighly correlated systems like the cuprates, this objectionwas not as serious, as an overall comparison of spectralintensities is not possible without detailed knowledge ofthe Raman scattering matrix elements given in Eq. (36).Good fits to the data of the A1g, B2g and B1g symmetries(with the overall intensities as free parameters) could beobtained (Devereaux and Einzel, 1995a). However, a sen-sitivity to band structure and higher harmonics of theenergy gap and Raman vertices found in the calculationsimplied a similar sensitivity to details of the materialswhich was not observed in the data.

Further, it was argued by Krantz and Cardona (1994)that scattering in a multi-band system - such as in theCuO2 bilayer - would yield a sharp intensity at twice thegap edge, which was again inconsistent with the data onsingle- and double-layer systems available at that time.

The issue of multiband scattering was partially re-solved by Devereaux et al. (1996). It was found that ifthe energy gaps were equal on at least two sheets of theFermi surface split by bi-layer hopping, a diverging in-tensity would be possible. However the intensity is onlyproportional to the difference of the individual Ramanvertices of the bands. This is a qualitative reason whythe multi-band case would give peaks only under quitespecial conditions, implying that the fits obtained fromthe single band case are still valid. Yet, it does not sat-isfactorily explain the sensitivity of the A1g response to

band and energy gap anisotropy factors.A recent undertaking to understand the origin of the

A1g intensity has involved the response of the peak topartial replacement (up to a few percent) of Cu by Znand/or Ni in YBa2Cu3O7−δ (Gallais et al., 2002; LeTacon et al., 2005; Limonov et al., 2002b; Martinho et al.,2004). Martinho et al. (2004) found that the intensity ofthe A1g peak was insensitive to either Ni or Zn doping,while the peak in B1g is suppressed by Zn, in agreementwith earlier results (Limonov et al., 2002b). As shownin Figure 36, Gallais et al. (2002) found that the peakposition in A1g was reduced by Ni impurities and madethe important observation that it followed that of themagnetic spin resonance mode (Sidis et al., 2004). LeTacon et al. (2005) found that Zn doping increased thelow frequency spectral weight and suppressed the B1g

peak without changing its position. It is not clear whythe position of the B1g peak does not change while Tc

decreases. Possible explanations are an inhomogeneousdistribution of the impurities or an accidental cancella-tion effect between the decreasing 2∆(Tc) and an increaseof the peak energy ΩB1g

peak because of defects (Devereaux,1995b). The shift of the A1g peak with Zn argued for atwo-component picture including a contribution from acollective mode such as the π resonance in the spin wavespectrum (Hinkov et al., 2004).

Two approaches concerning the presence of a collectivemode in the spectra superimposed upon a well-screenedbackground have been put forward recently by Venturini

80 9030

35

40

65

70

ERS B1g

peak ERS A

1g peak

INS resonance

Ene

rgy

(m

eV)

Tc (K)

FIG. 36 Raman and neutron-resonance peak ener-gies as a function of the critical temperature Tc inYBa2(Cu1−xZnx)3O6.95 for x = 0, 0.01, and 0.03. The hor-izontal line for the B1g peak positions is just a guide to theeye, while the A1g peak and the neutron resonance are fit-ted by a straight line representing 5kBTc. From Gallais et al.(2002).

43

et al. (2000) and Zeyher and Greco (2002). Venturiniet al. (2000) showed that a collective mode, uniquely ap-pearing in A1g and not other channels due to symmetry,emerges due to coupling to the 41 meV spin resonancemode seen in neutron scattering measurements in a num-ber of compounds (Sidis et al., 2004). Zeyher and Greco(2002) argued that the peak in the B1g channel may beidentified as a collective mode split off from 2∆ (Blum-berg et al., 1997a) as a consequence of the simultaneouspresence of long-range d-CDW and d-superconducting or-der and that the A1g peak may be related to a supercon-ducting amplitude mode. In either case the sensitivity toparameters characterizing the anisotropies of energy gap,band structure, and Raman vertices were not present, yetit is still unclear which, if either, are able to explain theA1g peak in the superconducting state.

Yet a remarkable set of experiments by Goncharov andStruzhkin (2003) plotted in Figure 37 highlights some

Pressure/GPa0 5 10 15 20

Wav

enu

mb

er/c

m-1

200

300

400

A1g

Phonon

2∆/k

Tc

4

5

6

7

B1g

B1g

A1g

n-nopt, holes /CuO2

-0.08 0.00 0.08

2∆/k

Tc

4

8

12Bi2Sr2CaCu2O2+x

YBa2Cu3O7+x

FIG. 37 Pressure dependences of the superconducting peaksin YBa2Cu3O6.95 for B1g (full circles) and A1g (open circles)symmetries. The position of the B1g phonon is also indicated(triangles). The inset shows a comparison to the Raman dataon Bi2Sr2CaCu2O8+δ samples at ambient pressure with dif-ferent doping levels (Kendziora and Rosenberg, 1995). FromGoncharov and Struzhkin (2003). Reproduced with permis-sion from Wiley c© 2003.

shortcomings in the above mentioned scenarios. Uponpressure the peak in the B1g channel and Tc move down-ward in a similar fashion as upon overdoping (discussedin detail in Section IV.C), and the B1g phonon hardensas expected, while the peak position in A1g remains re-markably constant. This implies that the B1g peak isintimately tied to superconductivity and the A1g peakhas a substantial contribution from a channel which isrelatively insensitive to pressure and changes in super-conducting properties. In the discussions of possible can-didates for the mode highlighted above, the pressure sen-sitivity of spin, charge, and superconducting order wouldall be expected to be large. Thus the origin of this peakremains presently unclear. One can speculate that it maybe related to possible phonon modes involving mixed Bi-O and Ba-O which have been indicated to be less sensitiveto pressure, but clearly further work is needed to clarifythis matter (Goncharov and Struzhkin, 2003).

In multi-layer compounds with more than 2 adjacentCuO2 planes, a strong A1g peak in the superconductingstate occurs at roughly the peak frequency of the B1g

channel, as shown in Figure 38 for Bi2Sr2Ca2Cu3O10+δ.In addition, strong phonon resonances were found hereand for multilayer Hg compounds (Hadjiev et al., 1998),where some A1g c-axis phonons shift by as much as 20cm−1 at the superconducting transition, more than inany other high Tc compound, implying a strong renor-malization of the A1g continuum below Tc.

Recently, Munzar and Cardona (2003) argued that ac−axis plasma mode would be expected to be Raman ac-tive in cuprate materials with more than 2 CuO2 planes,giving rise to an additional contribution in the A1g chan-nel only due to mass fluctuations with opposite sign ondifferent CuO2 layers. The position of the plasma reso-nance was predicted to lie in a frequency range close tothe continuum peak, as shown in Figure 38.

3. Resonance Effects

As another experimental knob to turn, resonance stud-ies of the peaks developing in the superconducting stateprovide information on the character and the interactionsof the quasiparticles forming the condensate. For thisreason, studies of the gap feature as a function of theincoming photon energy were pursued. Effects in theenergy range of the gap were observed in various com-pounds with one (Blumberg et al., 2002; Kang et al.,1996) more than one CuO2 layer (Budelmann et al., 2005;Hadjiev et al., 1998; Limonov et al., 2002a; Rubhausenet al., 1999; Sacuto et al., 1998). There are two dis-tinct, though not necessarily independent, effects: (i)the spectral weight and the shape of some phonons ofpredominantly A1g symmetry change more or less dra-matically below Tc (Hadjiev et al., 1998; Limonov et al.,2002a), and (ii) the intensity of the electronic continuumis amplified in the vicinity of 2∆max (Fig. 38) (Blumberget al., 2002; Limonov et al., 2002a; Rubhausen et al.,

44

FIG. 38 Resonance enhancement of the of the A1g Ramanresponse in Bi2Sr2Ca2Cu3O10+δ at 10 K. From Limonov et al.(2002a).

1999). The resonance effects typically occur for excita-tion energies of 2 ± 0.2 eV and are particularly strongin compounds with 3 or 4 CuO2 layers (Limonov et al.,2002a).23 Importantly, the normal state is only weaklyaffected. Hence, although the resonance energy is close tothe 1.6 eV absorption edge (Singley et al., 2001; Uchidaet al., 1991), constraints are imposed on explanationsin terms of interband transitions between the lower andthe upper Hubbard band (c.f., Eq. (44) and Shastry andShraiman (1990)), since the width of the resonance ismuch larger than 2∆max. At least in the triple-layer com-pounds, an additional channel can be opened up belowTc due to the c-axis plasma resonance (Munzar and Car-dona, 2003) which may interact with low-energy phononsand renormalize their shape and intensity substantially.The existence of a resonating continuum in double-layercompounds is still a matter of debate. Some authorsfind the pair-breaking peaks to resonate (Blumberg et al.,1997a; Budelmann et al., 2005; Rubhausen et al., 1999),while others do not (Limonov et al., 2002a; Venturiniet al., 2002c).

Given this background, the strong resonance effectsin electron-doped Nd2−xCexCuO4 were a quite interest-ing and somewhat unexpected feature (Blumberg et al.,2002). While the B2g response in the superconductingstate is strongly enhanced toward the red, neither theB1g spectra below Tc nor the normal state spectra ingeneral are particularly sensitive to the energy of the ex-citing light. An explanation in terms of Hubbard physicsis certainly tempting but, for the symmetry and tem-perature dependence, not completely exhaustive. In ouropinion, the subject needs further experimental and theo-

23 It is interesting to note that the resonance of the two-magnonpeak is close to the charge transfer gap of 2.5 eV or higher (Blum-berg et al., 1997a; Knoll et al., 1996; Rubhausen et al., 1997).

retical clarification before arriving at a level of predictivepower.

After the observation of resonance effects inNd2−xCexCuO4 and along with an improved mate-rial quality the number of studies in electron-dopedsystems increased continuously and facilitated severalinteresting insights which will be summarized in thefollowing paragraph.

4. Electron versus Hole-Doped Materials

Some cuprates such as Nd2−xCexCuO4 crystallize inthe T ′ structure which is characterized by missing oxygenoctahedra (see Figure 32) and, as a consequence, a shortc-axis (Tokura et al., 1989). The maximal Tc of approx-imately 30 K is obtained in thin films of La2−xCexCuO4

(Naito and Hepp, 2000). None of the electron-dopedcuprates is completely ordered, and oxygen appearing inan apex position is the main defect (Radaelli et al., 1994)even in superconducting samples.

Phonons (Heyen et al., 1991) and crystal-field excita-

0 500 500 500 50

X 0.2

0 500

6

12

18

24

X 0.4

0

1

2

3

4

x 0.5

0

5

10

15

20

25

0

5

0

1

0

0

0.135+ 0 . 0 0 31 6 . 5

+ 1f i l m

Raman Shift (cm

-1 )

Ram

an R

esponse (

rel. u

nits)

B

2g B1g A1g

Pr

2-x

Ce

x

CuO

4-0.147+ 0.00323.5

+ 1.5. crystal

0.165+ 0.005

15+ 2crystalx

T

c(K)0.17+ 0.00513

+ 2crystal

0.17

+ 0.00213+ 0.5film

0.18

+ 0.00210+ 0.5filmxUND OPT OVD OVD OVD OVD

FIG. 39 Doping dependence of the low energy electronic Ra-man response of Pr

2 x

Ce

x

CuO

4

single crystals and thin filmsfor B2 g

, B1 g

and A

1 g

channels obtained with 647 nm exci-tation. The columns are arranged from left to right in orderof increasing cerium doping. Abbreviations UND, OPT andOVD refer to under-doped, optimally doped and over-dopedsamples, respectively. The normal state response (light/red)measured just above the respective T

c

is decomposed for theB2 g and B

1 g

channels into a Drude-like component with aconstant carrier lifetime (green dotted line) and an extendedcontinuum (yellow dotted line). Superconducting spectra(dark/blue) are taken at T ≈ 4 K. For the OPT crystal alow-frequency ω

3

power law is shown at B

1 g

symmetry. FromQazilbash et al. (2005).

45

tions (Jandl et al., 1993) in Nd2−xCexCuO4 were stud-ied soon after the discovery (Tokura et al., 1989) andexplained thoroughly. An exception was the A1g line at590 cm−1, assigned by Heyen et al. (1991) as a localizedmode of interstitial oxygen in apex position (cf. Fig. 32).Later on, Onose et al. (1999) could indeed demonstratethat the mode is suppressed after annealing the samplesin reducing atmosphere in order to induce or enhance Tc.The first electronic Raman spectra in the superconduct-ing state of slightly overdoped Nd1.84Ce0.16CuO4 showeda small gap anisotropy (Stadlober et al., 1995). The ratio4.1 ≤ 2∆/kBTc ≤ 4.9 is similar to that of strong-couplingconventional superconductors like Pb, Nb or Nb3Sn (seeTable II). The temperature dependence of the gap isrelatively close to the BCS prediction (Blumberg et al.,2002; Stadlober et al., 1995). This motivated an inter-pretation in terms of an anisotropic s-wave gap, yieldinga reasonable description of both the shape and the posi-tions of the B1g and B2g pair-breaking peaks (Stadloberet al., 1995).

Similar spectral shapes were also found at other dop-ing levels for both Nd2−xCexCuO4 and Pr2−xCexCuO4,as shown in Figure 39. The low-energy sides of the peakswere found to be almost doping independent and closerto those of hole-doped cuprates than to those in conven-tional superconductors when samples and instrumenta-tion facilitated improved measurements close to Ω = 0.

Recent ARPES (Armitage et al., 2001; Matsui et al.,2005) and interferometric experiments (Chesca et al.,2003; Tsuei and Kirtley, 2000) provided evidence of ad-type gap bringing antiferromagnetic spin fluctuationsback into play as a possible coupling mechanism. Sincethe diameter of the Fermi surface encircling (π, π) issmaller here than in hole-doped systems, the antiferro-magnetic ordering vector QAF = (π, π) connects spotsclose to (π/2, π/2) rather than in the vicinity of (π, 0)for p doping and the gap is expected to have a maximumat the hot spots and to decrease towards (π, 0).

On this basis Blumberg et al. (2002) proposed a noveltype of non-monotonic d-wave gap with maxima only ap-proximately 15 away from the diagonal. An explicitcalculation using the suggested ∆(k) (Blumberg et al.,2002) in a one-band model approximately reproduces theoverall line shapes but reveals discrepancies in the peakpositions (Venturini et al., 2003) (see also Blumberg et al.(2003)). Better agreement between experiment and the-ory can be obtained with a monotonic d−wave gap in atwo band picture (Liu et al., 2005) for which evidence hasbeen found by magnetotransport (Fournier et al., 1997).

Four remarks are to be considered: (i) In ARPES, thegap maximum is found approximately at the hot spotwhere a pseudogap is observed above Tc (Matsui et al.,2005). (ii) No quasiparticle peaks indicating coherence inthe superconducting state are resolved in ARPES (Ar-mitage et al., 2001; Matsui et al., 2005). (iii) Twicethe gap energy observed by ARPES is ∼ 40% smallerthan that derived from the Raman spectra (Matsui et al.,2005). (iv) Phase-sensitive experiments (Alff et al., 1999)

and results on the magnetic penetration depth (Kokaleset al., 2000; Skinta et al., 2002a) are supportive of ans-type gap.

Hence in spite of mounting evidence of d-wave pair-ing, explanations for the symmetry dependence of theRaman spectra and several other experiments are stillmissing. A possible explanation could lie in a crossoverfrom d to s pairing upon increasing doping (Skinta et al.,2002b). However, Raman data from differently dopedPr2−xCexCuO4 (see Figure 39) do not show a variationof the lineshape and sufficiently strong symmetry depen-dence of the pair-breaking features in the proper dopingrange (Qazilbash et al., 2005).

In the hole-doped systems, on the other hand, strongvariations of the lineshapes at A1g and B1g symmetry arefound although there is little doubt about the persistenceof the (dominant) d-wave nature of the superconduct-ing gap in the entire phase diagram (Tsuei and Kirtley,2000). The doping and polarization dependence of theRaman spectra in superconducting p-type cuprates willbe the subject of the next section.

C. Superconducting Gap: Doping Dependence

One of the early Raman experiments on doping effectsin overdoped Bi2Sr2CaCu2O8+δ showed the energy of thepair-breaking peak in B1g symmetry to decrease muchfaster than those at the other symmetries and Tc (Stauferet al., 1992). The first systematic study was performed byKendziora and Rosenberg (1995) on Bi2Sr2CaCu2O8+δ,shown in the inset of Figure 37. The doping level in thismaterial can be varied continuously and reliably between0.15 ≤ p ≤ 0.23 by changing the oxygen concentration(Kendziora et al., 1993; Triscone et al., 1991). At loweroxygen doping, the structure is possibly metastable orunstable, at higher doping the oxygen diffuses out evenat room temperature. Underdoping is better achievedby replacing Ca with Y. As an essential result, the pair-breaking peaks at A1g, B1g, and B2g symmetry are foundto depend in different ways on doping and, consequently,on Tc. This doping dependence is shown in Figure 40for recent measurements on Bi2Sr2Ca1−xYxCu2O8+δ bySugai et al. (2003). Most remarkably, the B1g pair-breaking peak is proportional to (1−p) rather than Tc inthe doping range indicated, but fades in intensity for un-derdoped materials. The peak energies in B2g symmetryhowever scale more or less with the transition temper-ature. For p < 0.15 the B1g peak becomes very weak,yet a clear 2∆ peak survives in the B2g channel for alldoping levels.

In the decade to follow there were numer-ous studies on differently doped cuprates, in-cluding Bi2Sr2CaCu2O8+δ,24 YBa2Cu3O6+x,25

24 (Blumberg et al., 1997a; Budelmann et al., 2005; Hackl et al.,

46

La2−xSrxCuO4,26 HgBa2CuO4,27 HgBa2Ca2Cu3O8,28Tl2Ba2CuO6,29 Tl2Ba2CaCu2O8,30 andTl2Ba2Ca2Cu3O10

31. A compilation of experimentalresults is shown in Fig. 41. The scatter of the datapoints partially reflects the development of the samplequality. However, there are also discrepancies betweenthe results which are related to the interpretation of theexperiments.

FIG. 40 Doping dependence of the B1g and B2g Raman spec-tra in Bi-2212 at 5 K and 100 K. From Sugai et al. (2003).

1996; Hewitt and Irwin, 2002; Liu et al., 1999; Opel et al., 2000;Rubhausen et al., 1999; Sugai and Hosokawa, 2000; Sugai et al.,2003; Venturini et al., 2003)

25 (Altendorf et al., 1992; Chen et al., 1993; Cooper et al., 1988b;Limonov et al., 1998, 2000; Masui et al., 2005; Nemetschek et al.,1997; Opel et al., 2000; Reznik et al., 1993; Sugai et al., 2003)

26 (Chen et al., 1994b; Naeini et al., 1999; Venturini et al., 2002a)27 (Gallais et al., 2005; Le Tacon et al., 2005)28 (Sacuto et al., 1998, 2000)29 (Blumberg et al., 1997b; Gasparov et al., 1997; Kang et al., 1996;

Nemetschek et al., 1993)30 (Kang et al., 1997)31 (Gasparov et al., 1997; Stadlober et al., 1995)

We first summarize the generally accepted featuresclose to and above optimal doping:

• at p 0.16 the peaks in the three Raman activesymmetries are in relative positions expected ford wave pairing, the ratio 2∆max/kBTc is approxi-mately 8,32

• in the overdoped range, p > 0.16, the B1g

peak frequency decreases faster than Tc obeyingΩB1g

peak/Tmaxc ≈ 46(0.28 − p) (with ΩB1g

peak and Tmaxc

in cm−1 and K, respectively),33

FIG. 41 Compilation of the position Ωpeak of the peak inthe Raman response in the superconducting state normal-ized to Tmax

c . B1g (open diamonds) and B2g (squares) ori-entations are shown for a variety of compounds. The re-spective references are color coded as follows: Bi-2212 [red(Venturini et al., 2002c), green (Sugai and Hosokawa, 2000;Sugai et al., 2003), cyan (Kendziora and Rosenberg, 1995),yellow (Blumberg et al., 1997a; Liu et al., 1999), pink (Hewittand Irwin, 2002), and gold (Masui et al., 2003)], LSCO [blue(Sugai et al., 2003)], Bi-2223 [grey (Masui et al., 2003)], Tl-2201 [purple (Gasparov et al., 1997, 1998; Kang et al., 1997;Nemetschek et al., 1993)], Tl-2223 [olive (Hoffmann et al.,1994; Stadlober et al., 1994)], and Hg-1201 [black (Gallais etal., 2005)]. The dashed black line is the interpolation formula6Tc/Tmax

c = 6[1 − 82.6(p − 0.16)2]. For comparison, resultsfor twice the maximal leading edge gap of ARPES (circles)(Campuzano et al., 2002) and for the peak-to-peak energy inthe tunneling density of states (triangles) (Zasadzinski et al.,2002) are included.

32 (Chen et al., 1994b; Cooper et al., 1988b; Devereaux et al., 1994a;Gasparov et al., 1997; Hackl et al., 1988b; Yamanaka et al., 1988)

33 (Blumberg et al., 1997a; Kendziora and Rosenberg, 1995; Masuiet al., 2003; Naeini et al., 1999; Sugai et al., 2003; Venturiniet al., 2002c)

47

• whenever the B2g peak can be observed its maxi-mum follows Tc, ΩB2g

peak ∝ Tc,34

• the A1g peak frequency follows either the magnetic(π, π) mode35 or, in the case of resonantly enhancedlight scattering, ΩB1g

peak.36

The gap close to the node, which is projected out inB2g symmetry, is also found by penetration depth, lowbias STM, and the Nernst effect to more or less follow thetransition temperature for all doping levels (Deutscher,1999, 2005; Panagopoulos and Xiang, 1998; Xu et al.,2000). However it is inconsistent with peak-to-peakmeasurements of tunneling density of states(Zasadzinskiet al., 2002), the energy of the (π, 0) peak in the spectralfunction(Campuzano et al., 2002), and thermal conduc-tivity (Sutherland et al., 2005), which when interpretedin terms of d−wave quasiparticle picture (Durst and Lee,2000) indicate that the gap energy continues to increasewith decreased doping. This highlights one of the majorissues in the cuprates of whether the superconductingpairing energy rises with underdoping or follows Tc, andthis certainly requires further study.

In many experiments, a second energy scale is observedwhich varies approximately as p0 − p, similar to ΩB1g

peak.The majority of authors (Timusk and Statt, 1999) call ita pseudogap ∆∗ opening below the crossover temperatureT ∗ (see Fig. 32) with ∆∗ ∝ T ∗. The understanding of thepseudogap is a matter of intense research at present. Apossible relation to the B1g Raman data will be discussedbelow in section IV.C.

In the underdoped range, the interpretation is morecontroversial. The main issues are whether or not theB2g pair-breaking peak can be observed at all dopinglevels and how the B1g spectra evolve for p < 0.16.

The B2g problem seems to converge with the improve-ment of the sample quality. The cleaner the samples, theclearer the B2g pair-breaking peak, as expected theoreti-cally (Devereaux, 1995b). The problem can be visualizedin YBa2Cu3O6+x where oxygen tends to cluster and toform pinning centers if the Cu-O chains are not com-pletely filled (Erb et al., 1996b). However, in addition tofully oxygenated YBa2Cu3O7, partially ordered phasesexist for x = 0.5 with Tc 60 K and x = 0.353 withTc → 0 (Liang et al., 2000, 2002) where every secondand third chain is filled, respectively. Therefore, the max-ima are pronounced in YBa2Cu3O6.50 and YBa2Cu3O6.98

(Opel et al., 2000), while the peak is smeared out at op-timal doping, YBa2Cu3O6.93, and practically disappearsslightly below (Sugai et al., 2003). The problem of oxy-gen clustering exists also in Bi2Sr2CaCu2O8+δ, but the

34 (Gallais et al., 2005; Kendziora and Rosenberg, 1995; Misochkoet al., 1999; Opel et al., 2000; Venturini et al., 2002b,c)

35 (Gallais et al., 2002; Le Tacon et al., 2005)36 (Limonov et al., 2002a)

resulting potentials are weaker and essentially doping in-dependent, as can be seen directly from the comparison ofdifferent overdoped samples (Opel et al., 2000; Venturiniet al., 2002c) exhibiting nearly constant intensity of theB2g pair-breaking features. In a similar fashion, partialreplacement of Ca by Y does not create a strong impuritypotential either. This was directly shown by electron spinresonance (ESR) (Janossy et al., 2003) in YBa2Cu3O6.1,where Ca in place of Y is a very weak impurity whichdoes not localize carriers even at low temperature anddoping. We conclude that Y and O doping can be usedmore or less simultaneously in Bi2Sr2CaCu2O8+δ as longas Y is distributed statistically. The direct comparisonof the data presented in Fig. 40 with those of Opel et al.(2000) and Venturini et al. (2002c) indeed shows thatthe B2g pair-breaking peaks have doping and dopant (O,Y) independent intensities if a high sample quality canbe maintained. Under these conditions, the B2g maximaseem to exist at all doping levels and apparently scalewith Tc. Since in YBa2Cu3O6+x and La2−xSrxCuO4 dop-ing changes both carrier concentration p and mean freepath , the pair breaking peaks appear and disappeardepending on the doping-dependent order.

It is important to remember these considerationsfor the analysis of the B1g data. This is partic-ularly important when intensity issues are discussed.Therefore, we focus first on Y-under- and O-overdopedBi2Sr2CaCu2O8+δ (Fig. 40) to determine the intensityevolution of the B1g pair-breaking structure. It becomesvery weak right below optimal doping while no significantweakening of the B2g structures is observed. The gradualsuppression of the B1g coherence peaks in Raman goesalong with the disappearance of the 2∆ peaks in scanningtunneling microscopy (McElroy et al., 2003). This is cor-roborated by results in partially ordered YBa2Cu3O6.5,in La1.9Sr0.10CuO4, HgBa2CuO4, and HgBa2Ca2Cu3O8,where the B2g structures are well resolved, while in B1g

symmetry, no pair-breaking effect could be found (Gal-lais et al., 2005; Naeini et al., 1999; Opel et al., 2000;Venturini et al., 2002b).

When the peak is observed in B1g in the underdopedregion, as shown in Figure 41, its position does not trackTc and lies within the scatter of points or slightly belowthe values of 2∆ generally obtained via ARPES (Cam-puzano et al., 2002) and peak-to-peak positions in tun-neling measurements (Zasadzinski et al., 2002).37 It hasbeen argued by Chubukov et al. (1999, 2006) and Zeyherand Greco (2002) that the B1g peak in Raman measure-ments in underdoped compounds may be due to a collec-tive mode appearing below Tc from either spin-coupling

37 We point out however that there is considerable uncertainty indetermining the gap from ARPES, as both the leading edge andthe peak of the spectrum have been used. In either case, the anti-nodal spectral function is very broad in underdoped systems andidentifying an energy scale from a broad feature is not withoutuncertainty.

48

or d−CDW order, respectively, split off from twice thegap maximum 2∆. Yet this interpretation is inconsistentwith the doping behavior of the pairing gap determinedfrom B2g symmetry, as discussed above. In addition, thefindings of the doping behavior of the anti-nodal qua-siparticles in the normal state (listed below in SectionIV.C) indicate that anti-nodal quasiparticles have be-come incoherent already above optimal doping. There-fore it is not intuitively obvious how a propagating modecould emerge deep in a region of incoherence. An alter-native scenario is that the peak observed in B1g channelsis strongly altered by incoherence and is a measure of thebinding energy of localized anti-nodal electrons, such asvia the formation of resonance-valence bond singlets orsmall polarons. Yet why a peak should appear remainsto be understood. In any case, this issue highlights an-other vexing problem in the cuprates: how involved arethe anti-nodal electrons in superconductivity for under-doped systems.

There were reports about B1g pair-breaking peaks atlow doping p ≤ 0.1 (Blumberg et al., 1997a; Slakey et al.,1990b) appearing already above Tc, at a doping indepen-dent position of approximately 600 cm−1. In the discus-sion of the existence of preformed pairs in the pseudo-gap state above Tc and of collective modes inside thegap (see Section II.D.7) that are indicative of the pair-ing potential, this result has obviously some importance.However, in the vast majority of the experiments, the ob-servation could not be confirmed. Further complicationcomes from the existence of an oxygen impurity mode inclose vicinity (Hewitt et al., 1999; Quilty et al., 1998).Future studies have to clarify this issue.

In conclusion, Raman scattering experiments revealtwo energy scales in the superconducting state whichhave a different dependence on doping: while the peaksin B2g symmetry follow the transition temperature, themaximum energy of those in B1g symmetry decreases as(p0 − p). In the majority of the experiments, the co-herence peaks in B1g symmetry are found to fade awayrapidly for p < 0.16. The origin of the features observedat small doping remains controversial.

D. Normal State: Dichotomy of Nodal and Anti-NodalElectrons

The studies of the symmetry and doping dependence ofthe Raman spectra in the superconducting state are sug-gestive of interactions with a pronounced structure in mo-mentum space which, in addition, vary with doping (see,e.g., Fig. 41). One of the key questions is therefore howthe interactions renormalize the quasiparticles and theirresponse functions in the normal state.38 Owing to the

38 The properties of the normal state have been studied and re-viewed extensively in the last two decades. Recent reviews are

crystal structure of the cuprates, the anisotropy of the in-plane to the out-of-plane transport translates into a mo-mentum dependence of the in-plane transport properties(Devereaux, 2003b; Forro, 1993; Turlakov and Leggett,2001). Hence, Raman scattering can substantially sup-plement optical conductivity measurements above Tc. Ananisotropy of the normal-state Raman spectra was actu-ally observed soon after the gap anisotropy.39

1. Unconventional Metal-Insulator Transition

The momentum dependence of the transport as mea-sured by Raman scattering is indeed dramatic. Gener-ally, the nodal region along the diagonal of the Brillouinzone (B2g symmetry) remain essentially unchanged atall doping levels, while the (π, 0) regions (B1g symme-try) suffer an overall loss of oscillator strength by ap-proximately an order of magnitude if the doping level isreduced from 0.22 to 0.10 (Fig. 42). When this dopingdependence was first observed, Katsufuji et al. (1994) re-alized that the variation of the B1g spectra with p cannotbe explained by only changing the filling of a rigid bandwhere the ratio of the B1g to the B2g scattering intensi-

FIG. 42 Direct comparison of the low-energy B1g and B2g Ra-man response functions measured at temperatures indicated(all below Tc) in La2−xSrxCuO4 for different values of Sr dop-ing. The scale is the same for all frames. From Naeini et al.(1999).

by Timusk and Statt (1999), Loram and Tallon (2001) and Basovand Timusk (2005). A thorough study of the dc and Hall trans-port properties in various compounds performed on high-qualitysamples of the last generation was presented recently by Andoet al. (2004)

39 (Blumberg et al., 1999; Hackl et al., 1996; Katsufuji et al., 1994;Naeini et al., 1999; Opel et al., 2000; Reznik et al., 1993; Slakeyet al., 1991; Staufer et al., 1990; Yamanaka et al., 1996)

49

ties is essentially determined by (t/t′)2 with t and t′ thenearest and next-nearest neighbor hopping matrix ele-ments, respectively (Einzel and Hackl, 1996). Changingthe ratio (t/t′) enough to account for the observed Ra-man intensities leads to unrealistic band structures. Thesuggested importance of correlation effects on the Ramanintensities was pointed out early on.

The intensity changes of the B1g spectra as a func-tion of doping (Fig. 42) are accompanied by a qualitativechange of the temperature dependence at a given dopinglevel. For Bi2Sr2CaCu2O8+δ, the effect is shown in therange 0.15 ≤ p ≤ 0.23 (Fig. 43). For B2g symmetry, thereis little change of both the overall intensity and the ini-tial slope which depends on temperature as expected fora metal. As opposed to B2g symmetry, the temperaturedependence of the B1g spectra reverses sign (Fig. 43 (a)),indicating non-metallic behavior at p = 0.15. As outlinedabove (Eq. (55)), the initial slope of the response functionis proportional to a transport lifetime τ or, equivalently, aconductivity. Accordingly, the inverse of it, [∂χ′′

µ/∂Ω]−1

is a resistivity which, for the Raman selection rules, ismomentum sensitive as indicated by the index µ.

It has been shown by Opel et al. (2000) that both theenergy dependences and the magnitudes of Γµ(Ω, T ) =/τµ(Ω, T ) and m∗

µ(Ω, T )/mb = 1 + λµ(Ω, T ) can bedetermined from the spectra using a memory functionanalysis (Gotze and Wolfle, 1972) in combination withan energy integral over χ′′

µ/Ω (“sum rule”). By extrap-olating the results to Ω = 0, very reliable numbers forΓ(Ω → 0, T ) can be obtained.40

In Fig. 44, the “Raman resistivities” Γµ(Ω, T ) in thelimit Ω = 0 are plotted for µ = B1g, B2g at various tem-peratures and doping levels. The B2g results comparewell to conventional resistivities ρ(T ) using the Drudeexpression Γµ(Ω = 0, T ) = ε0(ωpl)2ρ(T ) with the renor-malized plasma frequency ωpl. With the plasma fre-quency in eV and the resistivity in µΩcm, one findsΓµ(T ) 1.08(Epl)2ρ(T ). At p = 0.23, the transportis essentially isotropic. Below p = 0.22, anisotropy de-velops quite abruptly, and for p < 0.16, the tempera-ture dependence of the ΓB1g (Ω = 0, T ) becomes non-metallic. This crossover behavior has been interpreted in

40 The memory function or extended Drude analysis is particularlyuseful for a single-component response. Further details and lim-itations are discussed in the appendix of reference (Opel et al.,2000). In Raman scattering the integral is as crucial as theplasma frequency in IR spectroscopy if magnitudes and energydependences are to be derived in a similar way as in the analysisof the reflectivity (Basov and Timusk, 2005). “Sum rule” mightbe somewhat misleading and should be used with care since theintegral over χ′′

µ(Ω)/Ω ∝ σ′(Ω) (see Eq. (6)) is not a conservedquantity like the number of carriers in the famous f-sum rule.Without the integral the energy dependence of one of the quan-tities must be dropped and/or a fit to model functions for Γµ(Ω)and 1 + λµ(Ω) is required (Blumberg et al., 1999; Hackl et al.,1996; Naeini et al., 1999; Slakey et al., 1991; Yamanaka et al.,1996).

terms of an unconventional metal-insulator transition at0.20 < p < 0.22, where the antinodal transport is grad-ually quenched while the nodal one remains essentiallyunaffected (Venturini et al., 2002a).

Since the current vertex has a similar k dependence asthe B2g Raman vertex, IR spectroscopy and, similarly,dc transport project out mainly the nodal part of theFermi surface (Devereaux, 2003b), and show thereforegood qualitative agreement with the B2g Raman results,which exhibit little variation with doping for 0.1 < p <0.23 beyond the change of ωpl, as shown in Figure 44.The suppression of the antinodal transport goes alongwith a reduction of the quasiparticle strength Zk in thevicinity of (π, 0) (Kim et al., 2003), and a collapse ofthe Korringa law (T1T )−1 = const as observed by NMR,where T1 is the spin-lattice relaxation time (Alloul et al.,1989; Billinge et al., 2003).

Below p 0.16, insulating behavior can also be ob-served in conventional transport at low temperature ifsuperconductivity is suppressed with high magnetic fields(Ando et al., 2004a; Boebinger et al., 1996). The loga-rithmic divergence of the resistivity in the limit T → 0indicates localization of the carriers. The crossover frommetallic to non-metallic behavior occurs at temperaturesT < Tc, well below those found in B1g Raman scatter-ing. This is consistent with the observation that the B2g

spectra do not show any anomaly in the normal state.The synopsis of NMR, conventional, and Raman trans-port leads to the quite consistent conclusion that carriersget gradually localized upon decreasing doping. Accord-ing to the Raman results, the suppression of free-carriertransport starts at p 0.21 at the anti-nodal regions ofthe Fermi surface and gradually moves towards the nodalpoints with decreasing p. The conductivity apparentlydisappears only very close to or at zero doping, makingconnection to recent ARPES results which show quasi-particles along the nodal direction even at the lowest (yetfinite) doping level (Yoshida et al., 2003).

Although there are no studies of both the doping and

X M

X M02468

10

02468

x 0.5

2gB

1gB

0 400 800 0 400 800 0 400 800

226K

105K196K

242K

91K190K

206K114K80K

2gB

1gB

2gB

1gB

Energy [cm-1]

//(

,T) [

cps/

mW

]

FIG. 43 Direct comparison of the low-energy B1g and B2g Ra-man response functions measured at temperatures indicated(all above Tc) in Bi2Sr2CaCu2O8+δ for different doping levelsp = 0.15, 0.20, and 0.22 (from left to right). From Venturiniet al. (2002a).

50

temperature dependences in n-doped cuprates, the com-parison of the low-temperature data of Pr2−xCexCu2O4

(Fig. 39) to those in Bi2Sr2CaCu2O8+δ (Fig. 40) orLa2−xSrxCu2O4 (Fig. 42) is worthwhile. In fact, a sig-nificant decrease of the overall B1g intensity for small nis revealed whereas the continuum at B2g symmetry isnot as doping independent as on the p-doped side, butrather becomes weaker towards lower doping. Ramanrelaxation rates have been determined explicitly onlyfor Nd1.85Ce0.15Cu2O4 (Koitzsch et al., 2003). Both inB1g and B2g symmetry, Raman “resistivities” are closeto those found by conventional transport, as shown inFigure 45. This is certainly at variance with the re-sults in hole-doped systems at comparable carrier con-centrations. From the viewpoint of Raman scattering,n-doped cuprates at optimal doping in the normal statelook more like over-doped p-type samples with isotropictransport properties. The interpretation is not clear atthe moment, but this similarity may be perhaps relatedto the rather involved band structure on the n-doped side(Fournier et al., 1997; Onose et al., 2004).

2. Quantum Critical Point(s)

The quite unusual transport properties describedabove are accompanied by various crossover phenomena,such as the opening of a pseudogap (Timusk and Statt,1999), the collapse of the Korringa law (Billinge et al.,2003), and several other intriguing observations (Loramand Tallon, 2001). Recent studies show pseudogap phe-nomena also in electron doped cuprates (Alff et al., 2003;Onose et al., 2001, 2004). All those anomalies fit intothe broader context of a quantum critical point (QCP)

0

500

1000

1500

2000

0 200 0 200 0 200 0 200 0 200

p = 0.10

Temperature (K)

p = 0.15 p = 0.16 p = 0.19

B1g

B2g

p = 0.23

Bi Sr CaCu O2 2 2 8+

0(T

)(c

m)

-1

X

M

2 g

B

X M 1 gB( a ) ( b ) ( c ) ( d ) ( e ) FIG.44Ramanrelaxationrates(•resistivitiesŽ)at B

1g

andB

2g symmetriesofBi

2

S r

2

CaCu

2

O 8+ . B

1g

andB 2gissensi-tivetoanti-nodalandnodalregionsoftheBrillouinzoneas i n d i c a t e d . T h e d a s h e d l i n e s r e p r e s e n t t r a n s p o r t d a t a . U s i n g

51

anen and Gunnarsson, 1989), charge ordering such asstripes or density waves (Andergassen et al., 2001; Castel-lani et al., 1995; Chakravarty et al., 2001; Kivelson et al.,2003) and Fermi surface deformation fluctuations (Met-zner et al., 2003). In all cases, static order is found only,if at all, at very low doping and/or in specifically modi-fied structures such as La2−y−xReySrxCuO4 for y 0.4and Re = Nd,Eu (Klauss et al., 2000; Tranquada et al.,1995) or the nickelates. In La1.775Sr0.225NiO4 (Pashke-vich et al., 2000) and La1.67Sr0.33NiO4 (Blumberg et al.,1998; Yamamoto et al., 1998) the formation of static dis-tortions due to the formation of stripes are big enoughto split phonon and magnon lines due to zone doublingand to suppress the B1g continuum at low energies.

Of course, static order is much easier to verify thanfluctuating order. Nevertheless, fluctuating incommen-surate spin order has been observed in La2−xSrxCuO4

for x ≥ 0.055 (Fujita et al., 2002), YBa2Cu3O6.85, andYBa2Cu3O6.6 (Hinkov et al., 2004), while commensu-rate spin order is seen at low energies in YBa2Cu3O6.353

(Stock et al., 2006). This means that in a temperaturedependent volume characterized by a coherence lengthξs(T ) a superstructure of the antiferromagnetically or-dered spins is established for higher doping levels. Atthe same time the charges which are expelled from theantiferromagnetic regions becoming spatially organizedas well. This type of order has similarity with a liquidcrystal and is sometimes referred to as nematic (Kivel-son et al., 1998, 2003). From these experiments it can-not be decided whether the fluctuating magnetic super-structure is a property of the spins or of the charges.For statically ordered La2−y−xNdySrxCuO4 with dopingx = 1/8, Tranquada et al. (1995) showed that charge or-dering precedes spin ordering when the temperature isreduced. It has indeed been shown that charge orderingfluctuations are quite common phenomena in correlatedsystems (Castellani et al., 1995; Metzner et al., 2003).

As long as there is no long range or static order whichcompetes with superconductivity, the fluctuations can es-tablish Cooper pairing (Chakravarty et al., 2001; Peraliet al., 1996). For this reason, the understanding of thedynamics of incipient charge and spin order is a very in-teresting subject in the cuprates.

3. Role of Fluctuations and Incipient Ordering Phenomena atsmall doping

The study of dynamical order requires inelastic probestypically sensitive at finite q. In the case of charge or-dering, resonant X-ray scattering is the most promisingas it can provide direct evidence for the charge order-ing at a particular wavevector (Abbamonte et al., 2004),while neutrons can be used only if the charges modulatethe lattice or the spin structure. While optical meth-ods are confined to q = 0 two excitations with oppo-site momenta can be created, such as multi-phonon ortwo magnon scattering. Therefore light can be scattered

from Cooper pairs or if an electron-hole pair couples totwo excitations. It depends on the individual contexthow fluctuation phenomena are best studied.

In the copper-oxygen systems, an unexpected addi-tional component in the B1g spectra was observed inLa1.90Sr0.10CuO4: as shown in Figure 46, a peak is foundin the 100 cm−1 range which gains intensity and moves tovery low but finite energy with decreasing temperature(Tassini et al., 2005; Venturini et al., 2002b).42 At highenergy for all temperatures, and at high temperature forall energies, the spectra of the cuprates with p 0.10in general are similar and show a strong anisotropybetween nodal and anti-nodal regions in the Brillouinzone (see Fig. 42, 43, and 44). At low temperaturethe low energy feature is unique to La1.90Sr0.10CuO4,while the spectra become flatter in YBa2Cu3O6.5 andBi2Sr2(Y0.38Ca0.62)Cu2O8+δ (see Fig. 43). To appreci-ate the differences, one has to recall that the conventionaltransport properties of all these compounds are very sim-ilar (Ando et al., 2004). Before we discuss possible inter-

305 K255 K182 K88 K

188 K124 K65 K36 K

x = 0.02 x = 0.10

(a) (b)

200 4000

0

1

2

3

4

5

6

600 600

B2gB1g

Raman shift (cm )-1

’’(

,T)

(cps/m

W)

200 4000

La Sr CuO2-x x 4

FIG. 46 Low energy response of underdoped La2−xSrxCuO4.A Drude-like peak (Zawadowski and Cardona, 1990) with acharacteristic energy Ωc(x, T ) is revealed after subtraction ofthe 2D response of the CuO2 planes. At x = 0.02 (a) and0.10 (b) the additional response is observed in B2g and B1g

symmetry, respectively. The styles of the lines (colours) donot correspond to similar temperatures but rather highlightthe scaling of the response with temperature: similar spectraare obtained if the temperatures differ by approximately afactor of 2. From Tassini et al. (2005).

42 We note that the the lines observed recently by Gozar et al.(2004) in La2−xSrxCuO4 (x = 0, 0.01) have nothing in com-mon with the response discussed here and were clearly identi-fied as one-magnon excitations (see also Silva Neto and Benfatto(2005)).

52

pretations we wish to compare results for different dopinglevels and materials.

From the comparison of the low energy peak to in-frared results, a relationship to fluctuating stripe or-der was conjectured (Venturini et al., 2002b). For fur-ther support, it seems worthwhile to exploit the selectionrules Raman scattering offers. The effect in B1g symme-try is indeed compatible with the orientation of stripesalong the Cu–O bonds at p = x = 0.10 (Fujita et al.,2002). This is simply because order along the principalaxes of an essentially tetragonal lattice corresponds toan orthorhombic distortion. Fluctuations correspond tomicro-twinning, meaning that x and y cannot be accessedindividually. Hence, B1g symmetry measuring only thedifference xx − yy (on a microscopic scale) projects outexactly this distortion. As a familiar example, we re-call that fully oxygenated YBa2Cu3O7 has Cu–O chainsalong the crystallographic b axis making b > a. Even ifthe sample is twinned, the chain contributions are alwayssuperimposed on the B1g spectra, while the xx and yyspectra are equal (as opposed to a single-domain crystal).For x < 0.055, a reorientation by 45 is observed by neu-tron scattering (Fujita et al., 2002). Since B1g and B2g

are equivalent modulo a π/4 rotation in the basal plane ofa tetragonal lattice, the related peak should now appearin B2g rather than in B1g symmetry. In fact, this hasbe observed in La1.98Sr0.02CuO4, as shown in Figure 46(Tassini et al., 2005).

0 200 400 6000

1

2

3

4

0

1

2

3

4

5

''(

,T

)(c

ps/

mW

)

energy (cm )-1

328 K283 K190 K86 K11 K

Y123

B1g

B2g

p = 0.02

(a)

(b)

FIG. 47 Raman response χ′′µ(ω, T ) of

(Y0.97Ca0.03)Ba2Cu3O6.1 in B1g (a) and B2g (b) sym-metry. The doping level is close to p = 0.02. From Hacklet al. (2005).

Spectra similar to those in La1.98Sr0.02CuO4 (Fig. 46)are found in (Y0.97Ca0.03)Ba2Cu3O6.1 with p 0.02(Fig. 47 (b)). Owing to the homogeneity of the sam-ple, the peak is very narrow and well defined. As the lowenergy peak emerges, the intensity of the continuum issuppressed by roughly 30 % over an energy range of ap-proximately 550 cm−1 in a similar though much strongerway than at higher doping (Nemetschek et al., 1997; Opelet al., 2000). In B1g symmetry, there is an overall lossof spectral weight up to much higher energies, similarto YBa2Cu3O6+x and Bi2Sr2CaCu2O8+δ at higher dop-ings, demonstrating the transition to a correlated insula-tor (Freericks and Devereaux, 2001).

Simultaneously with the studies in La1.90Sr0.10CuO4,the quasi 1D ladder compound Sr14−xCaxCu24O41 wasinvestigated (Blumberg et al., 2002; Gozar et al., 2003)which is considered a model system for the high-Tc

cuprates (Dagotto and Rice, 1996; Dagotto, 1999; Sigristet al., 1994). At temperatures above approximately450 K low energy spectral peaks similar to those in

0

4 0

8 0

120

160

0 100 200 300 400 500 600 700

634

589

550

495 K

384 K

456 K

309 K

250 K210

Sr14

Cu24

O41

Ram

an r

espo

nse

(rel

. u.

)

Frequency (GHz)

0

2 0

4 0

6 0

0 5 1 0 1 5 2 0 2 5 Raman shift (cm-1) S r 2 Ca1 2 Cu2 4 O 4 1

6 1 5 6 K

5 4 1 6 K 4 9 6 K

4 5 9 6 K

3 1 0 K 2 6 7 2 2 2 1 7 0

0 2 0 4 0 6 0Raman shift (cm-1)

Ram

an r

espo

nse

Aω

ωΓΓ2 2+

a

c

(a)

(b)

FIG. 48 Temperature dependent Raman response for cc po-larization in ladder compounds. Upper inset: fit of the datawith Eq. 49 plus a small background. Lower inset: Two-legladder structure. From Gozar et al. (2003).

53

La1.90Sr0.10CuO4 are found (Fig. 48), although their en-ergies and widths are an order of magnitude smaller. Thewidth obeys an Arrhenius law with a doping independentactivation energy ∆ of approximately 2100 K. In the in-sulator (x = 0), the conductivity (for T < 300 K) revealsa similar ∆ while for x = 12 the conductivity is metallicabove 70 K (Gozar et al., 2003; Gozar and Blumberg,2005a). The position of the maximum and the width de-crease upon cooling. In both cases, for T < 450 K thepeaks move below the detection limit of 2 cm−1. Be-low 250 K, no indication of the peaks can be detectedany more. It is interesting to note that at least in un-doped Sr14Cu24O41 a charge-ordered state develops be-low TCO ≈ 250 K (Abbamonte et al., 2004).

In the ordered state Sr14Cu24O41 has an optical re-sponse which is well described by that of a pinned chargedensity wave (Blumberg et al., 2002; Littlewood, 1987).For T > 250 K, the results are interpreted in terms of adamped plasma oscillation above TCO (Blumberg et al.,2002; Gozar et al., 2003; Gozar and Blumberg, 2005a).Since free carriers damp the mode, the width is expectedto increase proportional the conductivity just opposite towhat one would expect from free carrier response. Thiscomplicates the interpretation of the results in metal-lic Sr2Ca12Cu24O41 where the width still increases withtemperature while the conductivity decreases. Gozaret al. (2003) argue that different regions of the Fermisurface contribute to the transport and the damping inessentially 2D Sr2Ca12Cu24O41.

A comparison of panels (a) and (b) of Figure 48 showsthat the continuum surviving at low temperature is sig-nificantly stronger in the metal (Fig. 48 (b)). We there-fore think that the low-energy mode found at T > 250 Kcould also originate from incipient charge order. InSr14−xCaxCu24O41 the mode is then superimposed on afree electron response in a similar way as in the high-Tc

cuprates.It is interesting that similar types of spectra are also

found in Bi1−xCaxMnO3 (see section III.E), exhibitingcomparable temperature dependences. While the resultsin ladders are interpreted in terms of an overdampedplasma mode in a CDW system above the ordering tem-perature (Blumberg et al., 2002; Gozar et al., 2003; Gozarand Blumberg, 2005a), the response in the cuprates wasproposed to originate from fluctuations of the chargedensity in the vicinity of a charge-ordering instability(Caprara et al., 2005). Since the charge modulation ob-served e.g. by tunneling microscopy (Hoffman et al.,2002; Howald et al., 2003; Vershinin et al., 2004) is atfinite q, two fluctuations have to be exchanged to fulfillthe q = 0 selection rule in Raman scattering (Caprara etal., 2005; Venturini et al., 2000). Caprara et al. (2005)have shown that charge ordering fluctuations at incom-mensurate wavevectors determined via neutron scatter-ing (Fujita et al., 2002) yield proper lineshapes and se-lection rules in La1.90Sr0.10CuO4 and La1.98Sr0.02CuO4.To which extent the spin channel is involved and whethersimilar considerations apply to other dopings and cuprate

families have not been explored yet and remain impor-tant future topics.

In contrast to Sr14−xCaxCu24O41, the low energy peakis strongly doping dependent in La2−xSrxCuO4. Thetemperature scale is different by a factor of 2 for the twodoping levels studied (Fig. 46), implying that T ∗(0.02) ≈2T ∗(0.10). The doping level pc determined from extrap-olating T ∗ to zero, T ∗(pc) = 0, yields 0.15 ≤ pc ≤ 0.20in close vicinity of the QCP inferred from other exper-iments. It is therefore possible that the low energy re-sponse is related to the thermal and quantum fluctua-tions above a hidden critical point.

In summary, the comparison of the cuprates studiedsuggests that there is a superposition of two anomaliesin B1g symmetry: (i) below p 0.22 antinodal qua-siparticles become localized in all compounds as canbe observed at sufficiently high temperature; (ii) withdecreasing temperature a well-defined peak develops inLa1.90Sr0.10CuO4 at low energy. Although nothing com-parable can be resolved in Y- and Bi-based compounds atthe same doping level, the additional response may notnecessarily be absent. It rather can mean that the low en-ergy features are broader and shifted to higher energies,hence becoming very similar in shape to the responsefrom the 2D planes. Since the relationship between su-perconductivity and spin and/or charge fluctuations is akey issue in the cuprates further work seems worthwhilehere.

V. CONCLUSIONS AND OPEN QUESTIONS

The results discussed in the course of this reviewdemonstrate that Raman spectroscopy is and has beenan invaluable tool to investigate the dynamics of stronglycorrelated electrons. Improvements in the experimentaltechnique have opened up the field to a variety of newsystems. Light scattering has offered unique insights intodynamics in different regions of the Brillouin zone, show-ing the development of ordering and the competition be-tween various phases as the role of correlations increases.Materials such as superconductors with charge-densitywave order, correlated insulators, ruthenates, mangan-ites, and finally the superconducting cuprates show therich variety of phenomena which have been unveiled viaRaman investigations. Along the way, light scatteringhas considerably deepened our knowledge of dynamicsand correlation effects, and has provided several key in-gredients towards the development of a comprehensivetheoretical description of these materials.

Summarizing the findings on correlated systems overthe past several years, the key ingredients stemming fromRaman investigations include: (i) The development ofanisotropies as correlations are increased. Symmetry-selective measurements provide a tool to zoom in on thedynamics in different regions of the Brillouin zone. Thisis evidenced by Raman scattering studies on MgB2 aswell as the cuprates in the normal and superconducting

54

phases, and the ordered phases in the ruthenates andthe manganites. (ii) The existence of collective modes.Using Raman techniques the prevalence of certain typesof order as a consequence were identified. Evidence forthe importance of collective modes comes from CDW-superconductors and magnon scattering in antiferromag-netic insulators, and implications for a number of possiblefluctuation or ordering modes exist in the cuprates. (iii)A qualitative understanding of quantum critical behav-ior. The symmetry projection of parts of the Brillouinzone completes the picture of the battleground betweencompeting orders concomitant with an underlying quan-tum phase transition. Here, spectral weight transfers as afunction of temperature, doping, and pressure on a num-ber of materials combined with polarization dependentstudies have opened a new door in the area of quantumcriticality.

It is clear that a number of issues are still of primaryinterest in correlated systems in general. These include(i) the origin of the electronic continuum in a number ofcompounds which look surprisingly similar at first pass.(ii) The origin of the polarization dependence in mate-rials which exhibit instabilities towards ordered phases.(iii) The mapping of Brillouin zone-projected electron dy-namics close to a quantum critical point.

The cuprates continue to provide a wealth of infor-mation and puzzles in the area of superconductivity andstrong electronic correlations. Issues in which consen-sus has been reached include (i) the presence antiferro-magnetic correlations over a wide range of doping levelsevidenced from the two-magnon peak, (ii) the broad con-tinuum in the normal state as a common feature of allthe cuprates, (iii) the d−wave nature of the pair statebelow Tc in a number of hole-doped materials derivedfrom the low frequency power laws and polarization de-pendences, (iv) the dichotomy between the dynamics ofB1g and B2g quasiparticles at low dopings, where the lowfrequency anti-nodal B1g behavior is governed by inco-herence at the same time as the nodal B2g quasiparti-cles show relatively doping-independent metallic charac-ter, similar to the findings of transport quantities, and(v) the disappearance of this dichotomy for appreciablydoped samples.

In the cuprates there are still several unsettled ques-tions. (i) What is the origin of the A1g and/or B1g peaksin the superconducting state? Do they originate fromthe redistribution of superconducting quasiparticles orare they collective modes or of lattice origin? (ii) Howdoes the nodal/anti-nodal dichotomy picture of coher-ence/incoherence merge into metallic description at highdoping, BCS superconductivity for temperatures belowTc, and antiferromagnetism near half filling? (iii) Whatis the microscopic origin for low energy peaks at low dop-ing, and how from this can information be obtained onthe competition between ordered phases?

These are issues which we believe will form the plan ofdevelopment of Raman scattering in the cuprates as wellas other materials in the years to come. The continua-

tion of investigations on many new and well-characterizedsamples will further our knowledge of materials and cor-relations, and applications of the use of pressure andmagnetic field will allow an exploration of quantum crit-icality and evolution of anisotropic electron dynamics ina variety of systems. The rapid development of Ramanscattering as we have outlined indicate that the study ofthe electronic dynamics of complex materials will remaina vibrant and promising field of research.

Acknowledgements

We acknowledge support by the Alexander von Hum-boldt Foundation, ONR Grant No. N00014-05-1-0127,and NSERC (T.P.D.) and by the Deutsche Forschungs-gemeinschaft under grant nos. Ha 2071/2 and Ha 2071/3(R.H.). The latter project is part of Research Unit 538.

We would first like to express our gratitude to A. Za-wadowski who kindled and fostered our interest in thefield of Raman spectroscopy and whose guidance and sci-entific insight is immeasurably appreciated. We wouldalso like to express our thanks to the current and formermembers of our groups for their cooperation, daily dis-cussions, and many useful comments: Ch. Hartinger, B.Moritz, R. Nemetschek, M. Opel, W. Prestel, A. Seaman,B. Stadlober, L. Tassini, F. Venturini and F. Vernay. Wealso would like to take this opportunity to acknowledgeour collaborators in the field: Y. Ando, H. Berger, R.Bulla, A. Chubukov, S. L. Cooper, D. Einzel, A. Erb,L. Forro, J. Freericks, Y. Gallais, K. Hewitt, J. C. Ir-win, A. Janossy, A. Kampf, M. V. Klein, K. Maki, J.Naeini, A. Sacuto, A. Shvaika, I. Tutto, A. Virosztek, andA. Zawadowski. Over the years, we have also benefitedtremendously from scientific discussions with many othercolleagues: Y. Ando, O. K. Andersen, K. Andres, D. Be-litz, L. Benfatto, G. Blumberg, A. Bock, S. V. Borisenko,N. Bontemps, I. Bozovic, P. Calvani, M. Cardona, B. S.Chandrasekhar, T. Cuk, A. Damascelli, C. Di Castro, J.Fink, P. Fulde, M. Gingras, A. Goncharov, A. Griffin,R. Gross, G. Guntherodt, W. Hanke, J. Hill, R. W. Hill,S. Ishihara, A. Janossy, M. Jarrell, K. Kamaras, H.-Y.Kee, B. Keimer, C. Kendziora, P. Knoll, M. Le Tacon,P. Lemmens, S. Maekawa, N. Mannella, D. Manske, I.Mazin, W. Metzner, G. Mihaly, L. Mihaly, N. Nagaosa,D. Pines, L. Pintschovius, D. Reznik, M. Rubhausen, T.Ruf, G. Sawatzky, R. Scalettar, Z.-X. Shen, E. Sherman,R. R. P. Singh, V. Struzhkin, S. Sugai, C. Thomsen, T.Tohyama, C. M. Varma, W. Weber, J. Zaanen and R.Zeyher. Lastly, it is a pleasure to thank S. L. Cooper, J.Freericks, M. Opel, I. Tutto, A. Shvaika, and A. Zawad-owski for the critical reading of this review article.

T.P.D. and R.H. would like to acknowledge the hos-pitality and support of the Walther Meissner Institutand the Laboratoire de Physique du Solide at the EcoleSuperieure de Physique et de Chimie Industrielles and,respectively, and the University of Waterloo, where partsof this review were completed.

55

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