Electrodynamics of Correlated Electron Materials

Electrodynamics of Correlated Electron Materials

Dimitri N. Basov Department of Physics, University of California San Diego, La Jolla, California 92093-0319, U.S.A.    Richard D. Averitt Department of Physics, Boston University, Boston, Massachusetts 02215, U.S.A.    Dirk van der Marel Départment de Physique de la Matière Condensée, Université de Genève, CH-1211 Genève 4, Switzerland    Martin Dressel 1. Physikalisches Institut, Universität Stuttgart, Pfaffenwaldring 57, 70550 Stuttgart, Germany    Kristjan Haule Department of Physics, Rutgers University, Piscataway, NJ 08854, USA
Abstract

We review studies of the electromagnetic response of various classes of correlated electron materials including transition metal oxides, organic and molecular conductors, intermetallic compounds with - and -electrons as well as magnetic semiconductors. Optical inquiry into correlations in all these diverse systems is enabled by experimental access to the fundamental characteristics of an ensemble of electrons including their self-energy and kinetic energy. Steady-state spectroscopy carried out over a broad range of frequencies from microwaves to UV light and fast optics time-resolved techniques provide complimentary prospectives on correlations. Because the theoretical understanding of strong correlations is still evolving, the review is focused on the analysis of the universal trends that are emerging out of a large body of experimental data augmented where possible with insights from numerical studies.

Contents

I Introduction

In their report on the Conference on the Conduction of Electricity in Solids held in Bristol in July 1937 Peierls and Mott wrote: “Considerable surprise was expressed by several speakers that in crystals such as NiO in which the -band of the metal atoms were incomplete, the potential barriers between the atoms should be high enough to reduce the conductivity by such an enormous factor as 10Mott and Peierls (1937). The “surprise” was quite understandable. The quantum mechanical description of electrons in solids - the band theory, developed in the late 1920-s Sommerfeld (1928); Bethe (1928); Bloch (1929) - offered a straightforward account for distinctions between insulators and metals. Furthermore, the band theory has elucidated why interactions between 10 cm electrons in simple metals can be readily neglected thus validating inferences of free electron models. According to the band theory NiO (along with many other transition metal oxides) are expected to be metals in conflict with experimental findings. The term “Mott insulator” was later coined to identify a class of solids violating the above fundamental expectations of band theory. Peierls and Mott continued their seminal 1937 report by stating that “a rather drastic modification of the present electron theory of metals would be necessary in order to take these facts into account” and proposed that such a modification must include Coulomb interactions between the electrons. Arguably, it was this brief paper that has launched systematic studies of interactions and correlations of electrons in solids. Ever since, the quest to fully understand correlated electrons has remained in the vanguard of condensed matter physics. More recent investigations showed that strong interactions are not specific to transition metal oxides. A variety of and electron intermetallic compounds as well as a number of -electron organic conductors also reveal correlations. In this review we attempt to analyze the rich physics of correlated electrons probed by optical methods focusing on common attributes revealed by diverse materials.

Central to the problem of strong correlations is an interplay between the itineracy of electrons in solids originating from wavefunction hybridization and localizing effects often rooted in electron-electron repulsion Millis (2004). Information on this interplay is encoded in experimental observables registering the electron motion in solids under the influence of the electric field. For that reason experimental and theoretical studies of the electromagnetic response are indispensable for the exploration of correlations. In Mott insulators Coulomb repulsion dominates over all other processes and blocks electron motion at low temperatures/energies. This behavior is readily detected in optical spectra revealing an energy gap in absorption. If a conducting state is induced in a Mott insulator by changes of temperature and/or doping, then optical experiments uncover stark departures from conventional free electron behavior.

Of particular interest is the kinetic energy of mobile electrons that can be experimentally determined from the sum rule analysis of optical data (Section II.4) and theoretically from band structure calculations. As a rule, experimental results for itinerant electronic systems are in good agreement with the band structure findings leading to in simple metals (see Fig.1). However, in correlated systems strong Coulomb interaction which has spin and orbital components Slater (1929) impedes the motion of electrons leading to the breakdown of the simple single-particle picture of transport. Thus interactions compete with itinerancy of electrons favoring their localization, and specifically suppress the value below unity (see Fig. 1). This latter aspect of correlated systems appears to be quite generic and in fact can be used as a working definition of correlated electron materials. Correlation effects are believed to be at the heart of many yet unsolved enigmas of contemporary physics including high-T superconductivity (Section V.1.1), the metal-insulator transition (Section IV, electronic phase separation (Section IV.6), and quantum criticality (Section III.5).

Optical methods are emerging as a primary probe of correlations. Apart from monitoring the kinetic energy, experimental studies of the electromagnetic response over a broad energy range (Section II.1) allow one to examine all essential energy scales in solids associated both with elementary excitations and collective modes (Section III). Complementary to this are insights inferred from time domain measurements allowing one to directly investigate dynamical properties of correlated matter (Section IV). For these reasons, optical studies have immensely advanced the physics of some of the most fascinating many-body phenomena in correlated electron systems.

Importantly, spectroscopic results provide an experimental foundation for tests of theoretical models. The complexity of the the problem of correlated electrons poses difficulties for the theoretical analysis of many of their properties. Significant progress has been recently achieved by computational techniques including the Dynamical Mean Field Theory (DMFT) offering in many cases an accurate perspective on the observed behavior (Section II.6). The ability of the DMFT formalism to produce characteristics that can be directly compared to spectroscopic observables is particularly relevant to the main topic of this review.

Figure 1: (Color online) The ratio of the experimental kinetic energy and the kinetic energy from band theory for various classes of correlated metals and also for conventional metals. The data points are offset in the vertical direction for clarity. Adapted from Qazilbash et al. (2009a).

In Fig. 2 we schematically show possible approaches towards an optical probe of interactions. It is instructive to start this discussion with a reference to Fermi liquids (left panels) where the role of interactions is reduced to mild corrections of susceptibilities of the free electron gas Mahan (2000). The complex optical conductivity of FL quasiparticles residing in a partially filled parabolic band is adequately described by the Drude model (see Subsection II.1 for the definition of the complex conductivity). The model prescribes the Lorentzian form of the real part of the conductivity associated with the intraband processesDrude (1900); Dressel and Grüner (2002); Dressel and Scheffler (2006):

(1)

where is the electronic charge, is the relevant density, and the band mass of the carriers which is generally different from the free electron mass , is the scattering rate and is the DC conductivity. In dirty metals impurities dominate and the scattering rate is independent of frequency thus obscuring the quadratic form of that is expected for electron-electron scattering of a Fermi liquid.111See, for example, Ashcroft and Mermin (1976); Pines and Nozières (1966); Abrikosov et al. (1963) Nevertheless, this latter behavior of has been confirmed at least for two elemental metals (Cr and -Ce) through optical experiments Basov et al. (2002); van der Eb et al. (2001) using the so-called extended Drude analysis (Section II.1). Another characteristic feature of Fermi liquids in the context of infrared data is that the relaxation rate of quasiparticles at finite energies is smaller than their energy: (at temperature ). The contribution of interband transitions is sketched in red and is usually adequately described through band structure calculations. The band structure results also accurately predict the electronic kinetic energy of a Fermi liquid that is proportional to the area under the intraband Drude contribution to the conductivity spectra (Section II.4).

Figure 2: (Color online) Schematic diagram revealing complimentary approaches to probing electronic correlations using IR and optical methods. Top panels show the momentum-resolved spectral function in a non-interacting metal (left), weakly interacting system (middle) and strongly correlated system (right). Characteristic forms of the real part of the conductivity , the frequency-dependent scattering rate and effective mass are displayed. The Drude intraband contribution to the conductivity (blue) develops a “side-band” in a system with strong electron-boson coupling. The corresponding enhancement of at energies below a characteristic bosonic mode can be registered through the extended Drude analysis (II.3). The magnitude of is related to the quasiparticle renormalization amplitude introduced in II.3. In a strongly correlated system (right panels) the oscillator strength of the entire intraband contribution is suppressed with the spectral weight transfer to the energy scale of the order of (light blue). The strength of this effect can be quantified through the ration of as in Fig.1 or equivalently through the ratio of optical and band mass: . Quite commonly this renormalization effect and strong electron-boson interaction act in concert yielding further enhancement of over the at .

One of the best understood examples of interactions is the Eliashberg theory of the electron-boson coupling Carbotte (1990). Interactions with a bosonic mode at modify the dispersion of electronic states near the Fermi energy (top panel in the middle row of Fig. 2). The spectra of reveal a threshold near reflecting an enhancement of the probability of scattering processes at . The spectral form of is modified as well, revealing a the development of a “side band” in at . However, the total spectral weight including the coherent Drude like structure and side bands (dark blue area in Fig.2) is nearly unaltered compared to a non-interacting system and these small changes are usually neglected. Thus, electron-boson interaction alone does not modify with respect to . Importantly, characteristic features of the bosonic spectrum can be extracted from the optical data Farnworth and Timusk (1974). Various analysis protocols employed for this extraction are reviewed in Section III.6. Coupling to other excitations, including magnetic resonances, also leads to the formation of side bands that in complex system may form a broad incoherent background in .

The right panels in Fig. 2 exemplify the characteristic electronic dispersion and typical forms of the optical functions for a correlated metal. Strong broadening of the dispersion away from indicates that the concept of weakly damped Landau quasiparticles may not be applicable to many correlated systems over the entire energy range. An optical counterpart of the broadened dispersion is the large values of . Finally, the low-energy spectral weight is significantly reduced compared to band structure expectations leading to that is substanitially less than unity. Suppression of the coherent Drude conductivity implies transfer of electronic spectral weight to energies of the order of intrasite Coulomb energy and/or energy scale of interband transitions. These effects are routinely found in doped Mott insulators for example (Section V) as well in other classes of correlated materials.222In transition metal oxides the magnitude of the onsite Coulomb repulsion can be both smaller or larger than the energy scale of interband transitions Zaanen et al. (1985). In organic conductors the hierarchy of energy scales is consistent with a sketch in Fig.2.

It is instructive to discuss dynamical properties of correlated electron systems in terms of the effective mass which in general is a tensor . For a general dispersion the mass is defined as , which reduces to a constant for free electrons with a parabolic dispersion. Deviations of from the free electron mass in simple metals are adequately described by band structure calculations yielding . This quantity is frequency independent (bottom left frame in Fig. 2) and enters the Drude equation for the complex conductivity Eq. (1). Electron-boson interaction leads to the enhancement of the effective mass compared to the band mass at as , quantifying the strength of the interaction (middle panel in the bottom row). The frequency dependence of can be evaluated from the effective Drude analysis of the optical constants. Strong electron-electron interaction can radically alter the entire dispersion so that is significantly enhanced over (right bottom panel). An equivalent statement is that is reduced compared to (see also Fig.1). Additionally, electron-boson interactions may be operational in concert with the correlations in modifying the dispersion at . In this latter case one finds the behavior schematically sketched in the bottom right panel of Fig. 2 with the thick red line.

Because multiple interactions play equally prominent roles in correlated systems the resulting many-body state reveals a delicate balance between localizing and delocalizing trends. This balance can be easily disturbed by minute changes in the chemical composition, temperature, applied pressure, electric and/or magnetic field. Thus correlated electron systems are prone to abrupt changes of properties under applied stimuli and reveal a myriad of phase transitions (Sections III, V). Quite commonly, it is energetically favorable for correlated materials to form spatially non-uniform electronic and/or magnetic states occurring on diverse length scales from atomic to mesoscopic. Real space inhomogeneities are difficult to investigate using optical techniques because of the fairly coarse spatial resolution imposed by the diffraction limit. Nevertheless, methods of near field sub-diffractional optics are appropriate for the task (Section V.2.1).

Our main objective in this review is to give a snapshot of recent developments in the studies of electrodynamics of correlated electron matter focusing primarily on works published over the last decade. Introductory sections of this article are followed by the discussion of excitations and collective effects (Section III) and metal-insulator transition physics (Section IV) exemplifying through optical properties these essential aspects of correlated electron phenomena. The second half of this review is arranged by specific classes of correlated systems for convenience of readers seeking a brief representation of optical effects in a particular type of correlated compounds. Given the abundant literature on the subject, this review is bound to be incomplete both in terms of topics covered and references cited. We conclude this account by outlining unresolved issues.

Ii Experimental Probes and Theoretical Background

ii.1 Steady-state spectroscopy

Optical spectroscopy carried out in the frequency domain from 1 meV to 10 eV has played a key role in establishing the present physical picture of semiconductors and Fermi liquid metals Dressel and Grüner (2002); Burch et al. (2008) and has immensely contributed to uncovering exotic properties of correlated materials Imada et al. (1998); Degiorgi (1999); Basov and Timusk (2005); Millis (2004). Spectroscopic measurements in the frequency domain allow one to evaluate the optical constants of materials that are introduced in the context of “materials parameters” in Maxwell’s equations. The optical conductivity is the linear response function relating the current to the applied electric field : . Another commonly employed notation is that of the complex dielectric function . The real and imaginary parts of these two sets of optical constants are related by and .333 In general higher-energy contributions from interband transitions (”bound charge” polarizability) are present apart from the quasi-free electrons that are summarized in replacing the factor 1 in this expression of . The static ”bound charge” polarizability is defined as the zero-frequency limit of , i.e. . Absorption mechanisms associated with various excitations and collective modes in solids (Fig. 3) give rise to additive contributions to spectra of and thus can be directly revealed through optical experiments. In anisotropic materials the complex optical constants acquire a tensor form. For instance, time reversal symmetry breaking by an applied magnetic field introduces non-diagonal components to these tensors implying interesting polarization effects Zvezdin and Kotov (1997). In the vast majority of optics literature it is assumed that the magnetic permeability of a material =1 with the exception of magnetic resonances usually occurring in microwave and very far infrared frequencies.444This common assertion has recently been challenged by the notion of “infrared and optical magnetism” Yen et al. (2004); Padilla et al. (2006); Shalaev (2007) realized primarily in lithographically prepared metamaterial structures but also in bulk colossal magneto-resistance manganites Pimenov et al. (2005, 2007). For inhomogeneous media, however, spatial dispersion becomes relevant that in general mixes electric and magnetic components Agranovich and Ginzburg (1984).

Figure 3: (Color online) Schematic representation of characteristic energy scales in correlated electron systems. These different processes give additive contributions to the dissipative parts of optical constants. TMO: transition metal oxides.

The complex optical constants can be inferred from one or several complementary procedures Dressel and Grüner (2002): (i) a combination of reflectance and transmission spectra obtained for transparent materials can be used to extract the dielectric function through analytic expressions, (ii) Kramers-Kronig (KK) analysis of for opaque systems or of for transparent systems, (iii) ellipsometric coefficients and for either transparent or opaque materials can be used to determine the dielectric function through analytic expressions,555This is straightforward only in the case of isotropic bulk materials; in the case of anisotropic materials or films some models have to be assumed. (iv) various interferometric approaches, in particular Mach-Zehnder interferometry, and (v) THz time-domain spectroscopy directly yield optical constants. These experimental techniques have been extensively applied to correlated matter. Extension of “optical” data to the microwave region is often desirable especially for superconductors and heavy electron materials which show interesting properties below 1 meV (Section VI).

ii.2 Pump probe spectroscopy

Ultrafast optical spectroscopy provides the possibility to temporally resolve phenomena in correlated electron matter at the fundamental timescales of atomic and electronic motion. Subpicosecond temporal resolution combined with spectral selectivity enables detailed studies of electronic, spin, and lattice dynamics, and crucially, the coupling between these degrees of freedom. In this sense, ultrafast optical spectroscopy complements time-integrated optical spectroscopy and offers unique possibilities to investigate correlated electron materials. This includes, as examples, phenomena such as electron-phonon coupling, charge-density wave dynamics, condensate recovery, and quasiparticle formation.

In time resolved optical experiments, a pump pulse photoexcites a sample initiating a dynamical response that is monitored with a time delayed probe pulse. Experiments on correlated electron materials fall into two categories as determined by the photoexcitation fluence Hilton et al. (2006). In the low-fluence regime (100  J/cm) it is desirable to perturb the sample as gently as possible to minimize the temperature increase. Examples of low fluence experiments discussed below include condensate dynamics in conventional and high-temperature superconductors (Sections III.7 and V.1.1, respectively), spin-lattice relaxation in manganties (Section V.3), and electron phonon coupling in Heavy-Fermions (Section V.2.2). At the other extreme are high-fluence non-perturbative experiments where goals include photoinducing phase transitions or creating non-thermally accessible metastable states having a well-defined order parameter.666Nasu (2004); Yonemitsu and Nasu (2008); Averitt and Taylor (2002); Kaindl and Averitt (2007); Kuwata-Gonokami and Koshihara (2006); Hilton et al. (2006). This is an emerging area of research that is quite unique to ultrafast optical spectroscopy. The coupling and interplay of correlated electron materials are of considerable interest in these high-fluence experiments as discussed in more detail in Sections IV.5 on photoinduced phase transitions and V.2.2 on the vanadates.

Low and high-fluence time-resolved experiments have been made possible by phenomenal advances in ultrashort optical pulse technology during the past fifteen years which have enabled the generation and detection of subpicosecond pulses from the far-infrared through the visible and into the x-ray region of the electromagnetic spectrum Kobayashi et al. (2005). Formally, ultrafast optical spectroscopy is a nonlinear optical technique. In the low-fluence regime, pump-probe experiments can be described in terms of the third order nonlinear susceptibility. However, more insight is often obtained by considering ultrafast optical spectroscopy as a modulation spectroscopy where the self-referencing probe beam measures the induced change in reflectivity or transmission Cardona (1969); Sun et al. (1993). This provides an important connection with time-integrated optical spectroscopy where the experimentally measured reflectivity and the extracted dielectric response are the starting point to interpret and analyze the results of measurements. Further, this is applicable to high-fluence experiments from the perspective of temporally resolving spectral weight transfer (Section II.4). In femtosecond experiments, the dynamics can be interpreted using the equation

(2)

where is the reflectivity, and are the induced changes in the real and imaginary parts of the dielectric function, respectively Sun et al. (1993). Insights into the electronic properties obtained from time-integrated measurements of (or the complex conductivity ) serve as a useful starting point to understand the quasiparticle dynamics measured using time-resolved techniques. Further, the development of time-gated detection techniques have enabled direct measurement of the electric field which, in turn, permits the determination of the temporal evolution of over the useful spectral bandwidth of the probe pulse.777See Averitt and Taylor (2002); Kaindl and Averitt (2007) and references therein for details.

The foundation for ultrafast experiments on correlated electron materials (at any fluence) is based on efforts during the past 25 years in understanding quasiparticle dynamics in semiconductors and metals.888Shah (1999); Chemla and Shah (2001); Allen (1987); Groeneveld et al. (1995); Sun et al. (1993); Axt and Kuhn (2004); Beaurepaire et al. (1996). In ultrafast optical experiments, an incident pump pulse perturbs (or prepares) a sample on a sub-100 fs time scale. This induced change is probed with a second ultrashort pulse that, depending on the wavelength and experimental setup, measures pump induced changes in the reflectivity, transmission, or conductivity. In the majority of experimental studies in condensed matter to date, the pump pulse creates a nonthermal electron distribution [Fig. 4 (1 2)] fast enough that, to first order, there is no coupling to other degrees of freedom. During the first 100 fs, the nonthermal (and potentially coherent) distribution relaxes primarily by electron-electron scattering [Fig. 4 (2 3)] Allen (1987); Fann et al. (1992); Groeneveld et al. (1995); Sun et al. (1993). Subsequently, the excited Fermi-Dirac distribution thermalizes through coupling to the other degrees of freedom (3 1).

Figure 4: Schematic description of dynamics in condensed matter probed with femtosecond spectroscopy. Prior to photoexcitation (1) the electrons, lattice, and spins are in thermal equilibrium. Photoexcitation creates (2) a nonthermal electron distribution. The initial relaxation proceeds primarily through electron-electron thermalization. Following thermalization, the electrons have excess energy which is transferred to other degrees of freedom on characteristic timescales ranging from approximately 1 ps for electron-phonon relaxation to tens of picoseconds for processes such as pair recovery across a gap. After Averitt and Taylor (2002).

There are of course, important aspects that Fig. 4 does not capture. Of particular importance are coherence effects where the impulsive nature of the initial photoexcitation leads to a phase-coherent collective response Shah (1999). This can include coherent phonons or magnons Thomsen et al. (1986); Dekorsy et al. (1996); Koopmans et al. (2005); Talbayev et al. (2005). However, even in the coherent limit, the results can often be interpreted as a dynamic modulation of the optical conductivity tensor though the connection with Raman scattering is important for certain experiments Merlin (1997); Misochko (2001).

An example that embodies what is possible with ultrafast optical spectroscopy we consider recent results on the formation of quasiparticles following above band-gap photoexcitation in undoped GaAs Huber et al. (2001).999These results provide a striking example of the onset of correlation following photoexcitation. In this experiment, pulses with 1.55 eV photon energy and approximately 10 fs duration excited an electron-hole plasma at a density of cm. Monitoring the dynamics requires probe pulses with sufficient temporal resolution and with a spectral bandwidth extending beyond 160 meV. This was achieved using a scheme based on difference-frequency generation in GaSe combined with ultrabroadband free-space electro-optic sampling Huber et al. (2001). The experimental results are displayed in Fig. 5, where spectra of the dynamic loss function are plotted at various delays between the optical pump and THz probe pulses. The imaginary part of is plotted in panel (a) and the real part in panel (b). This is a particularly useful form to display the data as it highlights what this experiment is actually measuring, namely, the evolution of particle interactions from a bare Coulomb potential to a screened interaction potential , where q is the momentum exchange between two particles during a collision.

In essence, becomes renormalized by the longitudinal dielectric function leading to a retarded response associated with the polarization cloud about the carriers. This is a many-body resonance at the plasma frequency where the loss function peaks at the plasma frequency with a width corresponding to the scattering rate. Thus, the results of Fig. 5 show the evolution from an uncorrelated plasma to a many-body state with a well-defined collective plasmon excitation. This is evident in Fig. 5(a) where, prior to photoexcitation, there is a well-defined peak at 36 meV corresponding to polar optical phonons. Following photoexcitation, a broad resonance appears at higher energies that evolves on a 100 fs timescale into a narrow plasma resonance centered at 60 meV. The response is described by the Drude model only at late delay times. These results are consistent with quantum kinetic theories describing nonequilibrium Coulomb scattering Huber et al. (2001).

Figure 5: Quasiparticle formation in GaAs at room temperature following excitation with 10 fs 1.55 eV pulses. The dynamic loss function is plotted as a function of frequency at various delays following photoexcitation. The response evolves to a coherent Drude response on a timescale of  175 fs as dressed quasiparticles are formed from an initially uncorrelated state at zero delay. From Huber et al. (2001).

In wide-band materials, it is possible to consider the dynamics largely in terms of the bandstructure where photoexcitation leads to changes in band occupation followed by subsequent relaxation processes. The example in Figure 5 is along these lines highlighting the dynamical evolution of low-energy spectral weight following photoexcitation of carriers across the badgap. In many ways, this can be considered as a model example of measurements in correlated electron materials in that it is the dynamical evolution of spectral weight (even if only over a narrow spectral range) that is monitored. The situation can be considerably more complicated in correlated materials starting with the fact that the electronic structure varies with occupancy. Thus, an excitation pulse can initiate a sequence of dynamical events quite different in comparison to the relaxation of a non-thermal electron distribution in a rigid band. For example, a change in orbital occupancy upon photoexcitation can near-instantaneously relax the need for a coherent lattice distortion (e.g. cooperative Jahn-Teller effect in the manganites) Tokura (2000); Polli et al. (2007) thereby launching a coherent lattice excitation that will in turn couple to other degrees a freedom. More generally, a delicate balance between various degrees of freedom occurs. Consequently, many such materials teeter on the edge of stability and exhibit gigantic susceptibilities to small external perturbations Dagotto (2003, 2005). Short optical pulses can play an important role as the external perturbation yielding a powerful tool to investigate dynamical interactions which determine the macroscopic response. Many examples will be encountered in the following sections.

The results presented in Fig. 5 represent the state-of-the art of what is currently feasible both in terms of experiment and theory of ultrafast optical spectroscopy as applied to condensed matter. The challenge is to utilize such experimental tools to investigate more complicated materials. This includes, as discussed in more detail below, the cuprates, manganites, heavy fermions, organics, and others. Interesting experimental insights have been obtained, but there is a need for theoretical studies focused on interpreting the results of time-resolved measurements. While theoretical studies on dynamics in wide-band semiconductors and heterostructures is relatively mature Axt and Kuhn (2004), to date, there have been relatively few theoretical studies on dynamics in correlated electron materials.101010For example, Unterhinninghofen et al. (2008); Howell et al. (2004); Takahashi et al. (2002); Ahn et al. (2004); Carbotte and Schachinger (2004). As described in this review, DMFT is a promising approach to analyze time-domain optical experiments and recently, a DMFT study along these lines has been published Eckstein and Kollar (2008).

ii.3 Theoretical Background

In an optical experiment a current is induced in the solid by the electric (proportional to where is the vector potential) and magnetic () components of the electromagnetic fieldCohen-Tannoudji (2004). The coupling in leading order of and is

(3)

The term of the interaction couples the angular momentum of the photon () to the orbital degree of freedom of the electron, leaving the spin unaffected. The term couples the photon angular momentum to the spin of the electron. In the absence of spin-orbit coupling these two couplings lead to the electric- and magnetic dipole selection rules respectively. Spin-orbit coupling relaxes these rules, which provides a channel for optically induced spin-flip processes through the term. Since this coupling contributes typically times the oscillator strength from the term, the latter coupling is usually neglected; here is the fine-structure constant. The optical conductivity is then computed by the linear response theoryMahan (2000)

(4)

where

(5)

is the current-current correlation function, and is the paramagnetic current density is temperature. Calculation of the current-current correlation function Eq. (5) requires the full solution of the many-body problem. Usually Eq. (5) is then expressed in terms of the one-particle Green’s function , the two particle vertex function and electron velocities by

(6)

as diagrammatically depicted in Fig. 6. All three quantities are matrices in the band index, i.e., , , and . The velocities are , where are a set of one particle basis functions.

Within a single band approximation, the Green’s function , the spectral function of electronic excitations and electronic self-energy are related by:

(7)

The self-energy in Eq.(7) contains information on all possible interactions of an electron with all other electrons in the system and the lattice. In the absence of interactions the spectral function is merely a -peak at whereas = =0. Interactions have a two-fold effect on the spectral function. First, the energy is shifted with respect to non-interacting case by the amount proportional . Second, the spectral function acquires a Lorentzian form with the width given by . The corresponding states are referred to as dressed states or quasiparticle states. The spectral function as well as complex self-energy are both experimentally accessible through photoemission measurements Damascelli et al. (2003).

Figure 6: (Color online) Diagrammatic representation of the current-current correlation function and the Bethe-Salpeter equation for the vertex correction to the optical conductivity.

Finally, the two particle vertex function (dark blue triangle in Fig. 6) can be computed from the fully irreducible two particle vertex function (light blue square in Fig. 6) through the Bethe-Salpeter equation depicted in the second line of Fig. 6. A consequence of this vertex is that an electron-hole pair can form a bound neutral particle, i.e. an exciton. In wide band insulators such as rocksalts Philipp and Ehrenreich (1963), semiconductors Klingshirn (1995), or organic materials Agranovich (2009), the exciton binding energies form a Rydberg series below the excition gap of unbound electron-hole pairs. In transition metal compounds the interaction is often strong enough to bind an electron-hole pair on a single atomic site (Section III.2).

ii.4 Sum rules

The response functions including optical constants of materials obey numerous sum rules Kubo (1957). The most frequently used sum rule is the f-sum rule for the real part of the optical conductivity :

(8)

This expression relates the integral of the dissipative part of the optical conductivity to the density of particles participating in optical absorption and their bare mass. The optical conductivity of a solid is dominated by the electronic response and therefore an integration of the data using Eq. (8) can be compared to the total number of electrons including both core and valence electrons.

A special role is played by the following sum rule for the optical conductivity of a single-band system governed by a Hamiltonian :

(9)

Here and the brackets denote the thermal average. In a tight binding model is the kinetic energyMaldague (1977); Baeriswyl et al. (1987):

(10)

where is the electron momentum distribution function. Since must accommodate the entire free carrier response (i.e. Drude peak and all side-bands due to interactions), one has to extend the integration to an energy above the free carrier response while still below the interband transitions. Kinetic energy Eq. 10 quantifies the oscillator strength of intraband transitions that can be equivalently characterized with the plasma frequency in weakly interacting systems or in a strongly interacting material where correlations renormalize the entire dispersion so that and .

Devreese et al. (1977) obtained the following “partial sum rules” for electrons occupying a band with a -independent mass, , coupled to phonons causing band mass to increase to the renormalized value m at energies below the phonon frequencies:

(11)
(12)

where is the narrow Drude peak alone whereas is the complete intraband contribution involving both the Drude peak and side-bands resulting from electron-phonon coupling (middle panels of Fig.2). A caveat: as pointed out above in many correlated electron materials the entire dispersion is modified by correlations leading to a suppression of the total intraband spectral weight Qazilbash et al. (2009a). This implies that in Eq.12 has to be replaced with higher optical mass and electron-boson mass renormalization is also executed with respect to , not . Following Maldague (1977) it is customary to define the effective spectral weight :

(13)

which has the meaning of the effective number of electrons contributing to electromagnetic absorption at frequencies below .

Of special significance for superconductors is the Ferrell-Glover-Tinkham (FGT) sum rule Tinkham (1996):

(14)

This equation relates the spectral weight “missing” from the real part of the conductivity upon transition to the superconducting state to the superfluid density which is proportional to the density of superconducting pairs and inversely related to their effective mass as: . Often, for practical reasons, the integration is limited to the free carrier response. Validity of the FGT sum-rule in this restricted sense requires that the electronic kinetic energy is unchanged below (see Section V.1.1 which discusses sum rule anomalies in high- cuprates). The superfluid density is of fundamental importance for the electrodynamics of superconductors. The sum rule [Eq. (14)] allows one to evaluate all three diagonal components of the superfluid density in anisotropic superconductors such as cuprates. 111111Basov et al. (1994b, 1995a); Liu and et al. (1999); Dulić et al. (2001); Dordevic et al. (2002); Homes et al. (2004a) and iron pnictidesLi et al. (2008a).

Experimental access to the quasiparticle kinetic energy is one of the important virtues of optical probes of correlations. An analysis of the one dimensional Hubbard Hamiltonian is particularly instructive in this regard Baeriswyl et al. (1986). Exact results obtained for a half-filled band reveal that the electronic kinetic energy is monotonically reduced with the increase of the on-site repulsion and tends to zero as . This result, along with the analysis of the spectral weight within the two-dimensional Hubbard model Millis and Coppersmith (1990), reinforces the notion that reported in Fig. 1 can be used as a quantitative measure of correlation effects.

Equation 9 is derived for a hypothetical single-band system where the kinetic energy may depend on temperature , magnetic field or other external stimuli. Strong variations of the electronic spectral weight commonly found in correlated electron systems upon changes of temperature or magnetic field may signal interesting kinetic energy effects. Consider, for example, a data set collected for a conducting system over the spectral range that is at least of the order of the width of the electronic band where the Fermi energy resides. The kinetic energy interpretation of Eq. (9) applied to such a data set is highly plausible. Quite commonly, one finds that the sum rule results in this case are temperature dependent Molegraaf et al. (2002); Ortolani et al. (2005). The only source of such a temperature dependence in a non-interacting system pertains to thermal smearing of the Fermi-Dirac distribution function leading to fairly weak effects scaling as Benfatto et al. (2005, 2006). In correlated electron systems this temperature dependence can become much more pronounced. This latter issue has been explicitly addressed within the framework of several scenarios of interacting electrons.121212Toschi et al. (2005); Benfatto et al. (2006); Norman et al. (2007); Karakozov and Maksimov (2006); Kuchinskii et al. (2008); Marsiglio (2006); Abanov and Chubukov (2004). We wish to pause here to strike a note of caution and stress that apart from intrinsic origins the variation of the electronic spectral weight may be caused by ambiguities with the choice of cut-off for integration of experimental spectra Benfatto and Sharapov (2006); Norman et al. (2007). Indeed, in many realistic materials including transition metal oxides intra- and inter-band contributions to the conductivity spectra commonly overlap unlike idealized schematics in Fig. 2.131313Examples of extensive experimental literature on sum rule anomalies in correlated systems can be found in the following references: Basov et al. (1999); Katz et al. (2000a); Basov et al. (2001); Molegraaf et al. (2002); Kuzmenko et al. (2003); Homes et al. (2004b); Santander-Syro et al. (2004); Boris et al. (2004); LaForge et al. (2008, 2009).

ii.5 Extended Drude formalism and infrared response of a Fermi liquid

In a conducting system physical processes responsible for renormalization of electronic lifetimes and effective masses also lead to deviations of the frequency dependent conductivity from conventional Drude theory. These deviations can be captured through the extended Drude formalismGötze and Wölfle (1972); Allen and Mikkelsen (1977):

(15)

The complex memory function has causal analytic properties and bears strong similarities with the electron self-energy for -points averaged over the Fermi surface. This analysis is particularly useful for the exploration of electron-boson coupling (Section III.6) and of power law behavior in quantum critical systems (Section III.5). The subtle differences between and the self-energy are discussed in a number of publicationsAllen (1971); Shulga et al. (1991); Dolgov and Shulga (1995).

In the absence of vertex corrections, the following approximate relation between of an isotropic Fermi liquid and the single particle self-energies was derived Allen (1971)

(16)

where is the self-energy of electrons with binding energy , and is the Fermi-Dirac distribution. The coupling of electrons to phonons or other bosonic fluctuations is described by the boson density of states multiplied with the square of the coupling constant, for phonons, for spin fluctuations, and in general. The self-energy is within this approximation

(17)

where the Kernel is a material independent function given by the Fermi and Bose distributions Allen (1971). In this set of expressions a double integral relates to and the optical conductivity, which is reduced to a single integral by the -reasonably accurate- Allen approximation Allen (1971); Shulga et al. (1991)

(18)

where is a material independent Kernel, different from . Marsiglio et al. (1998) derived in the limit of weak coupling and zero temperature

(19)

which for the optical spectra of KC Degiorgi et al. (1994c) resulted in the qualitatively correct electron-phonon spectral function.

If the low energy band structure can be approximated by a single “effective” band and the scattering is small, one may approximate the electron self-energy by a Fermi liquid expansion , with . Here is the quasiparticle renormalization amplitude. The low energy conductivity of such a Fermi liquid is given by

(20)

where is the non-interacting plasma frequency, is the non-interacting chemical potential, and is the regular part of the conductivity.

It is evident from Eq. (20) that the Drude weight is reduced by the quasiparticle renormalization amplitude , i.e., . Within the band structure method the Drude weight can be characterized by the effective density and the band mass by . The renormalized Drude weight, defined in Eq. (11), can be similarly expressed by . Hence the renormalized quasiparticle mass is . As expected, the quasiparticle dispersion , measured by ARPES, is also renormalized by the same amount.

The spectral form of the optical conductivity is usually more complicated then the Drude term alone and in addition contains both the incoherent spectral weight as well as many sidebands due to coupling to various excitations including magnetic and bosonic modes. These additional contributions are contained in . The plasma frequency is hence modified due to renormalization of quasiparticles and presence of other excitations by

(21)

where is the integral of the regular part of up to a cutoff . The cutoff should exclude the interband transitions, but should be large enough to include the intraband transitions of some low energy effective Hamiltonian. The total spectral weight , which is closely related to the kinetic energy of a corresponding low energy Hamiltonian, defines the optical effective mass via , as depicted by a dark blue area in Fig. 2. Hence the optical mass renormalization over the band mass is , which is smaller then the enhancement of the low energy quasiparticle mass , measured by ARPES. The optical mass enhancement is also sketcked in Fig. 2 as the high energy limit of the effective mass . The low energy quasiparticle effective mass is further enhanced by an amount . This additional enhancement can be obtained using the extended Drude analysis. Comparing Eq. (15) with Eq. (20) in the zero frequency limit, we see that . Hence the quasiparticle effective mass is

(22)

which is equal to the renormalization of the quasiparticle dispersion, as meassured by ARPES. Hence the optical effective mass of a correlated metal can be obtained from optical conductivity data by comparing the total spectral weight below some cutoff with the band structure method. To obtain the quasiparticle effective mass , one needs to further renormalize the mass by the factor , which can be obtained by the extended Drude model analysis.

Finally, for a very anisotropic Fermi liquid with strong variation of quasiparticle weight across the Fermi surface, the formula for the effective mass needs to be corrected. As shown by Stanescu et al. (2008), the quasiparticle effective mass measured by optics is roughly proportional to , where stands for the average over the Fermi surface. The effective mass measured by other probes can be different. In particular, the Hall effect experiments measure the effective mass proportional to , and the quantum oscillations experiments measure the effective mass proportional to Stanescu et al. (2008).

ii.6 Dynamical mean field theory

The theoretical modeling of correlated materials proved to be a very difficult challenge for condensed matter theorists due to the absence of a small parameter for a perturbative treatment of correlations, such as the small ratio between the correlation energy and the kinetic energy, or a small electron radius in the dense limit of the electron gas.

For realistic modeling of weakly correlated solids, the local density approximation (LDA) turn out to be remarkably successful in predicting the electronic band structure, as well as the optical constants. However LDA can not describe very narrow bands, found in many heavy fermion materials, nor Hubbard bands. Not surprisingly, it fails to predicts the insulating ground state in several Mott insulators and charge transfer insulators. The combination of LDA with static Hubbard correction, so called LDA+U Anisimov et al. (1991), was able to predict the proper insulating ground state in numerous correlated insulators. Being a static approximation, LDA+U works well for many correlated insulators with long range magnetic or orbital order. But the exaggerated tendency to spin and orbital order, the inability to describe the correlated metallic state, or capture the dynamic spectral weight transfer in correlated metals hindered the applicability of the method. A perturbative band structure method was developed over a course of several decades, named GW method Hedin (1965), and proved to be very useful for moderately correlated materials. In particular, its quasiparticle self-consistent version van Schilfgaarde et al. (2006) successfully predicted band gaps of several semiconductors. However, its perturbative treatment of correlations does not allow one to describe Mott insulators in paramagnetic state, nor strongly correlated metals.

The theoretical tools were considerably advanced in the last two decades mostly due to the development of the practical and powerful many body method, the dynamical mean field theory (DMFT) Georges et al. (1996). This technique is based on the one particle Green’s function and is unique in its ability to treat quasiparticle excitations and atomic-like excitations on the same footing. The dynamic transfer of spectral weight between the two is the driving force for the metal insulator transition in Hubbard-like models as well as in transition metal oxides.

Historically, it was not the photoemission, but optical conductivity measurements, in combination with theory Rozenberg et al. (1995a), that first unraveled the process of the temperature dependent spectral weight transfer. In these early days it was difficult to probe the bulk photoemission due to the issues with the surface states that precluded the detection of the the quasiparticle peak and its temperature dependence. On the other hand, the optical conductivity measurements on VO Rozenberg et al. (1995a) unambiguously proved that a small decrease in temperature results in a redistribution of the optical spectral weight from high energy (of the order of few eV) into the Drude peak and mid-infrared peak. It was nearly a decade later before photoemission Mo et al. (2003) detected the subtle effects of the spectral weight transfer between the quasiparticle peak and Hubbard band.

The accuracy of DMFT is based on the accuracy of the local approximation Georges et al. (1996) for the electron self-energy. It becomes exact in the limit of infinite lattice coordination (large dimension), and is very accurate in describing the properties of numerous three dimensional materials Kotliar et al. (2006).

Just as the Weiss mean field theory Weiss (1907) for an Ising model reduces the lattice problem to a problem of a spin in an effective magnetic field, the DMFT approximation reduces the lattice problem to a problem of a single atom embedded in a self-consistent electronic medium. The medium is a reservoir of non-interacting electrons that can be emitted or absorbed by the atom. The local description of a correlated solid in terms of an atom embedded in a medium of non-interacting electrons corresponds to the celebrated Anderson impurity model, but now with an additional self-consistency condition on the impurity hybridization Georges et al. (1996). The central quantity of DMFT, the one particle Green’s function, is thus identified as an impurity Green’s function of a self-consistent Anderson impurity problem. Diagrammatically, the DMFT approximation can be viewed as an approximation which sums up all local Feynman diagrams. Hence, the mapping to the Anderson impurity problem can be viewed as a trick to sum all local diagrams.

A second theoretical advance came when DMFT was combined with band structure methods Anisimov et al. (1997), such as the Local Density Approximation (LDA), in an approximation dubbed LDA+DMFT Kotliar et al. (2006). This method does not require one to build the low energy model to capture the essential degrees of freedom of a specific material, a step, which is often hard to achieve. In LDA+DMFT the extended and sometimes orbitals are treated at the LDA level, while for the most correlated orbital, either or , one adds to the LDA Kohn-Sham potential all local Feynman diagrams, the diagrams which start at the specific atom and end at the same atom Kotliar et al. (2006).

The LDA+DMFT approach allows one to compute both the one particle Green’s function and the current vertex entering Eq. (6) for the optical response. These quantities are normally expressed in the Kohn-Sham basis in which the one-particle part of the Hamiltonian is diagonal. The DMFT one-particle Green’s function (propagator in Fig. 6) in the Kohn-Sham basis is

(23)

where is the Kohn-Sham potential, and is the DMFT self-energy. The procedure of embedding the DMFT impurity self-energy to the Kohn-Sham basis was extensively discussed in Haule et al. (2010). Finally, the two particle vertex function (dark blue triangle in Fig. 6) can be computed from the fully irreducible two particle vertex function (light blue square in Fig. 6) through the Bethe-Salpeter equation depicted in the second line of Fig. 6. Within the DMFT approximation, the two particle irreducible vertex is local, i.e., does not depend on , or , and hence can be computed from the solution of the DMFT impurity problem Georges et al. (1996). It was first noticed by Khurana (1990), that the vertex corrections to the optical conductivity within DMFT approximation vanish in the one-band Hubbard-like model. This is because the electron velocity is an odd function of momentum , and does not depend on and , and hence the vertex corrections to conductivity vanish. In general, for multiband situations encountered in LDA+DMFT, the vertex corrections do not necessarily vanish even though the two particle irreducible vertex is purely local in this approximation. This is because, in general, velocities are not odd functions of momentum, which is easy to verify in the strict atomic limit. Nevertheless, the vertex corrections are small in many materials because they vanish at low energy, where a single band representation is possible, and are also likely sub-leading at intermediate and high energy, where the itinerant interband transitions dominate. To date, a careful study of the vertex correction effects within LDA+DMFT is lacking. In the context of the Hubbard model, Lin et al. (2009) demonstrated that vertex corrections substantially contribute to the optical conductivity at the scale of the Coulomb repulsion , whereas negligible contributions were found to the Drude and the mid-infrared peaks.

In the absence of vertex corrections, the optical constants, Eq. (4) on the real axis takes a simple form

(24)

where and the trace needs to be taken over all the bands Haule et al. (2005). Eq. (24) has been used in the majority of the LDA+DMFT calculations.

Iii Excitations and collective effects

iii.1 Free charge carriers

The electrical conduction of a material is governed by how freely charge carriers can move throughout it. In his seminal model, Drude (1900) considered the charge carriers to propagate independently. The span between two scattering events has an exponentially decaying probability characterized by the time and the mean free path . This scattering or relaxation time fully describes the dynamical response of the entire system to external an electric field, summarized in the complex frequency-dependent conductivity [Eq. (1)]. The Drude model does not take into account interactions with the underlying lattice, with electrons, or other quasi-particles. In his Fermi-liquid theory Landau (1956) includes electronic correlations, yielding an effective mass and also an effective scattering time Pines and Nozières (1966).

In heavy fermion materials the hybridization of nearly localized -shell electrons with quasi-free conduction electrons leads to an effective mass orders of magnitude larger than the bare electron mass Fisk et al. (1988); Grewe and Steglich (1991). Accordingly, the spectral weight [proportional to according to the sum rule Eq. (8)] and the scattering rate are significantly reduced Varma (1985a, b); Millis and Lee (1987); Millis et al. (1987). Hence, the charge carriers are extremely slow due to electron-electron interactions which shifts the relaxation rate into the microwave regime. As demonstrated in Fig. 7, Scheffler et al. (2005) probed the real and imaginary parts of the Drude response in UPdAl and UNiAl Scheffler et al. (2006, 2010) over three orders of magnitude in frequency and verified that the actual shape is perfectly described by Eq. (1), because impurity scattering still dominates over electron-electron scattering in spite of the strong renormalization.

Figure 7: (Color online) Optical conductivity spectrum (real and imaginary parts) of UPdAl at temperature  K. The fit by Eq. (1), with and  s, documents the excellent agreement of experimental data and the Drude prediction. The characteristic relaxation rate is marked by the decrease in and the maximum in around 3 GHz Scheffler et al. (2005).

More specific to the GHz range, Fermi-liquid theory predicts a renormalized frequency-dependent scattering rate Ashcroft and Mermin (1976); Pines and Nozières (1966); Abrikosov et al. (1963):

(25)

with the prefactors increasing as the square of the effective mass Kadowaki and Woods (1986), and depending on the material properties Rosch and Howell (2005); Rosch (2006). An experimental confirmation of relation (25) is still missing.

iii.2 Charge transfer and excitons

Optical transparency of insulating compounds is a consequence of the energy gap in the spectrum for electron-hole pair excitations, which, if final-state interactions between the electron and the hole can be neglected, corresponds to the gap between the valence and the conduction band. Different physical origins of the gap are known, and the corresponding insulators can be classified accordingly. For the purpose of the review we make a distinction between two main classes: (i) A gap caused by the periodic potential of the lattice. Standard semiconductors and insulating compounds fall in this class. (ii) A gap opened by on-site Coulomb repulsion (Hubbard ) on the transition metal ion with an odd number of electrons per site. A further distinction in the latter group is made according to the value of compared to the charge transfer energy needed for the excitation process where denotes a hole in the anion valence band Zaanen et al. (1985). When , processes of the type are the dominant charge fluctuation corresponding to the optical gap at an energy . On the other hand, when , corresponds to the optical gap at energy and fluctuations at an energy fall inside the interband transitions. The case corresponds to the limit of a Mott-Hubbard insulator, and is found on the left hand side of the series, i.e. vanadates and titanates, as well as organic compounds. The situation , indicated as ‘charge transfer insulator’ is common on the right hand side of the series; the cuprates and nickelates fall in this class. Coupling between different bands mixes the character of the bands on either side of the gap, which softens the transition from Mott-Hubbard insulator to charge transfer insulator as a function of . This is of particular relevance for substances with and of the same size, e.g. in Cr, Mn and Fe oxides Imada et al. (1998); Zaanen et al. (1985).

The Coulomb interaction can bind an electron and a hole to form an exciton, the energy of which is below the excitation threshold of unbound electron-hole pairs. This is illustrated by the example of cuprous oxide (CuO). This material is important in the quest for Bose-Einstein condensation of excitons Snoke et al. (1990), a goal which until now has remained elusive Denev and Snoke (2002). CuO is a conventional band-insulator with a zone center gap of 2.17 eV. The valence and conduction bands have the same (positive) parity at the zone-center, rendering direct transitions across the gap optically forbidden. The optical spectrum is therefore dominated by the , , , and exciton lines situated 2 to 22 meV below the gap. The excitonic ground state is split by the electron-hole exchange interaction into an optically forbidden singlet, and a triplet situated respectively 151 meV and 139 meV below the gap. The triplet corresponds to a weakly dipole allowed transition at 2.034 eV, whereas the singlet (2.022 eV) can be optically detected in a finite magnetic field Fishman et al. (2009). Detection schemes employing THz radiation generated by - transitions Huber et al. (2006) or THz absorption by - excitations Leinß et al. (2008); Fishman et al. (2006) of excitons created by laser excitation, allow to monitor the internal conversion of the excitons to the ground state as a function of time.

In organic molecular crystals electron-hole pairs can be bound on a single molecule. Due to the larger band mass as compared to typical semiconductors, the exciton binding energy is relatively large: In a two-photon absorption experiment Janner et al. (1995) the ground state exciton of C was observed at an energy 0.5 eV below the threshold of the electron-hole continuum at 2.3 eV.

Figure 8: (Color online) Absorption spectrum of CuGeO measured at 300 K for two different polarizations of the light. The bandgap energy is 3 eV. The peak at 1.75 eV is a phonon assisted copper exciton Bassi et al. (1996).

When a gap is opened by the on-site Coulomb repulsion, a special situation arises due to the fact that the energy of a charge-neutral local configuration change can be smaller than the correlation gap. The result is again an excitonic bound state below the electron-hole continuum. For example, in the spin-Peierls system CuGeO the upper Hubbard band is separated from the occupied oxygen states by a 3 eV correlation gap. Bassi et al. (1996) observed a Cu exciton at 1.75 eV (Fig. 8), far below the onset of the electron-hole continuum at 3 eV. This weak absorption is responsible for the transparent blue appearance of this compound. In the one-dimensional compound SrCuO Kim et al. (2008c) sharp peaks observed at 10 K were attributed to weakly bound excitons. The ground state in NiO is three-fold degenerate, and the remaining 42 states are spread over about 10 eV, grouped in 7 multiplets. About half of these are below the 4 eV correlation gap Sawatzky and Allen (1984). These excitons have been observed in optical absorption Newman and Chrenko (1959); Tsuboi and Kleemann (1994). In KCuF crystal field excitons were observed at 0.7, 1.05, 1.21, and 1.31 eV corresponding to a local d-d excitation from the groundstate to ,, and excited states Deisenhofer et al. (2008).

For LaCuO the electron-hole threshold is at 1.9 eV; Ellis et al. (2008) observed a crystal field exciton at 1.8 eV, as well as a peak at 2.2 eV which they attribute to a quasi-bound electron-hole pair occupying neighboring copper and oxygen atoms. YTiO (SmTiO) has a 0.6 eV Mott-Hubbard gap; Gossling et al. (2008) reported excitons corresponding to processus of the type on two neighboring Y-atoms, at 1.95 (1.8) eV, as well as other configurations at higher energies, having strongly temperature dependent spectral weight in the vicinity of the magnetic ordering transitions Kovaleva et al. (2007). Khaliullin et al. (2004) showed that, as a consequence of the temperature dependent orbital correlations, both superexchange and kinetic energy have strong temperature and polarization dependences, leading to the observed temperature dependence of the spectral weight.

iii.3 Polarons

Electron-phonon coupling quite generally renormalizes the mass, velocity, and scattering processes of an electron. The quasiparticles formed when phonons dress the bare electrons are referred to as polarons. However, different conditions in the solid require different theoretical approaches to the electron-phonon interaction. If the electron density is high, the Migdal-approximation holds and standard Holstein-Migdal-Eliashberg theory is applied Mahan (2000). Historically, the concept of a polaron started from the opposite limit, i.e. a low density electron system interacting strongly with lattice vibrations. In this case the starting point is that of individual polarons, out of which a collective state of matter emerges when the density of polarons is increased. In many ways a polaron is different from an undressed electron. The polaron mass is higher and the Fermi velocity lower compared to those of the original electron, and a phonon-mediated polaron-polaron interaction arises in addition to the Coulomb interaction.

The original description by Landau and Pekar considers that an electron polarizes the surrounding lattice, which in turn leads to an attractive potential for the electron Feynman (1955); Mahan (2000). The situation where the electron-phonon interaction is local, is decribed by the Holstein model Holstein (1959a, b). This potential is capable of trapping the electron, and a bound state is formed with binding energy . In the literature a distinction is usually made between large and small polarons. Both in the Holstein and the Fröhlich model the polaron diameter varies continuously from large to small as a function of the electron-phonon coupling parameter, but typically the Holstein (Fröhlich) model is used to describe small (large) polarons Alexandrov and Mott (1995). The Fröhlich model uses optical phonon parameters such as the longitudinal phonon frequency , which can be measured spectroscopically Calvani (2001). In transition metal oxides the dominant coupling is to an oxygen optical mode eV. The binding energy and mass enhancement factor in the weak and strong coupling limits are summarized in Table 1

weak coupling strong coupling
Table 1: Expressions for the Fröhlich coupling constant , polaron binding energy , and mass enhancement in the weak and strong coupling limits Alexandrov and Kornilovitch (1999).

where . In transition metal oxides the band mass is typically , and . The corresponding strong coupling values provide the upper limit for the binding energy ( eV) and the mass enhancement ().

In general, if the electrons interact with a single Einstein mode, the spectrum consists of a zero frequency mode and a series of sharp side-bands that describe the incoherent movement of a polaron assisted by phonons Devreese and Tempere (1998). In real solids these sharp side-bands are smeared out due to the fact that phonons form bands, and usually only the envelope function is expected Alexandrov and Bratkovsky (1999). In a pump-probe experiment it is possible to move the electron suddenly away from the centre of the surrounding lattice distortion. This sets up coherent lattice vibrations, which have recently been observed in GaAs using a midinfrared probe pulse Gaal (2007). Predictions of the energy of the midinfrared peak using the Fröhlich model are as high as in the strong coupling limit Myasnikova and Myasnikov (2008), and in the Holstein model Fratini and Ciuchi (2006). Consequently, in the case of the transition metal oxides, the Fröhlich coupling predicts a midinfrared peak at 0.7 eV at most.

Compound (eV) Ref. T increase
LaSrNiO 0.75 a weak redshift
FeO 0.6 b
LaSrNiO 0.5 c
PrSrMnO 0.5 d blueshift
BaKBiO 0.4 & 1.2 e no shift
LaSrMnO 0.4 f intensity-loss
NaVO 0.38 g no shift
LaCaTiO 0.31-0.38 h
LaTiO 0.31 i blueshift
VO 0.38 j blueshift
BiCaMnO 0.25 k intensity-loss
SrTiO 0.25 l blueshift
EuCaBaCuO 0.15 m
NdCuO 0.1 n
Table 2: Midinfrared peak positions for various compounds. a Jung et al. (2001), b Park et al. (1998), c Bi et al. (1993), d Jung et al. (2000b), ePuchkov et al. (1995); Ahmad and Uwe (2005), f Jung et al. (1999), g Presura et al. (2003), h Thirunavukkuarasu et al. (2006), i Kuntscher et al. (2003), j Baldassarre et al. (2007), k Liu et al. (1998), l van Mechelen et al. (2008), m Mishchenko et al. (2008), n Lupi et al. (1999).

If we now consider Table 2, we observe that in most cases the peak maximum is below 0.75 eV. An exception is formed by the high T superconductor BaKBiO where, in addition to a weaker peak between 0.33 and 0.45 eV, a strong peak has been observed at 1.2 eV. The latter peak was originally interpreted as a small-polaron mid-infrared peak Puchkov et al. (1995) and more recently as a purely electronic transition Ahmad and Uwe (2005). The formalism has been extended to arbitrary density of Fröhlich polarons by Tempere and Devreese (2001). By fitting a moderate electron phonon coupling (), they obtained an excellent agreement with the optical data for NdCuO Lupi et al. (1999). In contrast, the one-polaron model does not capture the optical line shape near the maximum of these data, despite the the very low doping level.

Electrons doped into the unoccupied Ti -band of SrTiO are believed to form polarons due to the Fröhlich-type electron-phonon coupling Eagles et al. (1995). Indeed, a midinfrared band characteristic of a polaron is observed at 0.25 eV Calvani et al. (1993); van Mechelen et al. (2008), which redshifts and splits when the temperature decreases (Fig. 9). The free carrier mass derived from the Drude spectral weight is implying moderate electron-phonon coupling and large Fröhlich polarons in this material.

Figure 9: (Color online) Optical conductivity of SrTiNbO for 0.01, 0.002, 0.009 and 0.02 at 300 K (top) and 7 K (bottom) van Mechelen et al. (2008). The broad, temperature dependent, mid infrared band between 100 and 750 meV corresponds to (multi-) phonon sidebands of the Drude peak. The narrow Drude peak contains approximately the same amount of spectral weight as the sidebands, implying that .

A clear trend in Table 2 is the large values of in the transition metal oxides containing Ni, Mn, or Fe, i.e. materials where a transition metal has an open shell with more than one electron or hole. Recent LDA calculations of the electron-phonon coupling strength of YBaCuO Heid et al. (2009) gave =0.26,0.27, and 0.23 along the , and -axis respectively. Addressing the problem of a single hole doped in the antiferromagnetic insulator, Cappelluti et al. (2007) and Mishchenko et al. (2008) argued that the electron-phonon and exchange coupling conspire to self-trap a polaron. Adopting they predicted a double structure in the midinfrared similar to the experimental data, i.e. a phonon sideband at 0.1 eV and a sideband at 0.5 eV of mixed phonon-magnon character. In a similar way, the high energy of the midinfrared peak of the transition metal oxides in the top of Table 2 may be a consequence of the combination of electron-phonon coupling and magnetic correlation.

iii.4 Optical excitation of magnons

In correlated electron systems the spin degrees of freedom are revealed by the collective modes emanating from the inter- electronic correlations. Depending on the state of matter, these modes can take the form of paramagnons for a regular metal Monthoux et al. (2007), spinons in the Luttinger liquid Giamarchi (2004a), triplons in spin-dimers Giamarchi et al. (2008), triplet excitons in insulators (see subsection III.2), or magnons in a ferromagnetic or antiferromagnetic state.

Ferromagnetic resonance (FMR) or antiferromagnetic resonance (AFR) occurs by virtue of coupling of the electromagnetic field to zone-center magnons. If inversion symmetry is not broken, the only coupling to electromagnetic field arises from the term in Eq. (3). The selection rules are then those of a magnetic dipole transition. Hence the resonance features are present in the magnetic permeability , while being absent from the optical conductivity . Asymmetry of the crystalline electric field upon the spins causes the AFR frequency to be finite even at , the interaction occurring via the spin-orbit coupling. AFR and FMR allow to measure magnetocrystalline anisotropy and spin-wave damping in the hydrodynamic limit Heinrich and Cochran (1993). Langner et al. (2009) have recently applied this technique to SrRuO, and demonstrated that the AFR frequency and its damping coefficient are significantly larger than observed in transition-metal ferromagnets. Technological advances using synchrotron sources permit to measure the absorption spectra as a function of magnetic field for  cm and fields up to 14 Tesla. The high sensitivity of this technique has led to the discovery of a novel, strongly field and temperature dependent mode in LaMnO Mihály et al. (2004). Sensitive detection of FMR by the time-resolved magnetooptic Kerr effect measures the time-evolution of the magnetization following an optical pump pulse Hiebert et al. (1997).

Optical single-magnon excitations arise not exclusively from the coupling: spin-orbit interaction allows photons to couple to magnons through the term of Eq. (3). Activation of this type of optical process requires the breaking of inversion symmetry, which is present in multiferroic materials due to their ferro-electric polarization (section V.5). The optical excitation of a single magnon can be explained if the coupling to the electric field is an effective operator of Dzyaloshinski-Moriya symmetry Cépas et al. (2001). In the ordered spin state one of the two magnons in the Hamiltonian is replaced by the static modulation of spin density. In cases where magnons are electric dipole active, this has important consequences: Optical phonons and single magnon waves of the same symmetry will mix. Moreover, two-magnon and single magnons can be excited by the electric field component of electromagnetic radiation Katsura et al. (2007).

An excitation at 44.5 cm was observed by van Loosdrecht et al. (1996) in the infrared transmission spectrum of the spin-Peierls phase of CuGeO. The observed Zeeman splitting identified it as a magnetic excitation Uhrig (1997). However, the selection rules are those of an electric dipole Damascelli et al. (1997a). Extensive magnetic field studies of the infrared spectra of NaVO, SrCu(BO) and SrCuO indicated mixing of phonon and magnon excitations in these compounds Rõõm et al. (2004b, a); Hüvonen et al. (2007). For these examples a dynamical Dzyaloshinski-Moriya coupling has been proposed by Cépas and Ziman (2004).

The first optical spectra of double magnon excitations were reported by Silvera and Halley (1966) in FeF, and interpreted by Tanabe et al. (1965) as the coupling of the electric field vector to the effective transition dipole moment associated with a pair of magnons. The coupling is nonzero only in the absence of a center of symmetry between the two neighboring spins, as is indeed the case in FeF rutile crystals. If the crystal lattice itself is centro-symmetric, electronic charge (dis)-order can still provide the inversion symmetry breaking field: In the quarter-filled ladder compound NaVO the “charged” magnon effect Popova et al. (1997); Damascelli et al. (1998), lent support to a symmetry breaking charge ordering transition at 34 K. Later investigations favored a zigzag type of charge order without the required inversion symmetry breaking. An alternative mechanism proposed by Mostovoy requires a dynamically fluctuating symmetry breaking field rather than a static one. In this process the photon excites simultaneously one low-energy exciton and two spinons Mostovoy et al. (2002).

The inversion symmetry is not broken by pairs of Ni ions in NiO. Yet, Newman and Chrenko (1959) reported a magnetic absorption at 0.24 eV. Mizuno and Koide (1964) attributed this to the simultaneous excitation of two magnons and an optical phonon, a process which is allowed by the electric dipole selection rules.

Figure 10: Evolution of the optical conductivity from weakly coupled chains via two-leg ladders to 2D layers at  K. (Top) of CaCuO for (solid line), DMRG result (circles) for and = 1300 cm. (Middle) of LaCaCuO for E (solid), DMRG calculation (closed symbols), for , = 0.2 and  cm Nunner et al. (2002). (Bottom) of the 2D bilayer YBaCuO for (solid). In a bilayer, the two-magnon contribution from spin-wave theory (dashed) contains an in- plane part (dotted) and an interplane part (dash-dotted). Here, the in-plane exchange is J = 780 cm and the inter- plane exchange amounts to Grüninger et al. (2000). The two- magnon peak corresponds to 2.88 for , and to 2.73 for Grüninger et al. (2003).

Strong renewed interest in the magnetic fluctuations in transition metal oxides was revived following the discovery of high superconductivity in cuprates. The observation of a peak at 0.4 eV in the anti-ferromagnetic Mott insulater LaCuO by Perkins et al., was initially interpreted as an intra-atomic exciton on the copper site Perkins et al. (1993). However, the lower bound of the excitons was expected at about twice that energy based on microscopic calculations Eskes et al. (1990); McMahan et al. (1990), as was confirmed by resononant inelastic x-ray scattering experiments Kuiper et al. (1998). Lorenzana and Sawatzky (1995a, b) therefore postulated that the 0.4 eV peak is due to a phonon assisted two-magnon process similar to NiO Mizuno and Koide (1964), and developed a theory for the optical conductivity spectra. This interpretation was confirmed by the excellent agreement between the experimentally observed optical spectra and the two-magnon+phonon model for moments in two dimensions Grüninger et al. (2000); Struzhkin et al. (2000). The line shape of the phonon-assisted two-magnon optical absorption of the 1D spin chain CaCuO Suzuura et al. (1996) is very well described by the two-spinon continuum Lorenzana and Eder (1997). In the ladder system LaCaCuOVuletić et al. (2006) the spectrum of the on-rung triplet bound state was found in perfect agreement with the theory of two-triplon excitations, and it allowed the precise determination of the cyclic exchange constant Windt et al. (2001); Nunner et al. (2002). The importance of quantum corrections to the linear spin-wave theory are illustrated by the comparison in Fig. 10 of the two-magnon plus phonon optical absorption spectra of chains, ladders and 2D planes with dynamical mean field renormalization group (DMRG) calculations and linear spin wave theory Grüninger et al. (2003). The multi-magnon excitations in the lower panel (YBaCuO), having energies exceeding , are clearly not captured by linear spin wave theory, an aspect which DMRG theory describes rather well as is demonstrated by the upper two panels.

iii.5 Power law behavior of optical constants and quantum criticality

In certain materials a quantum phase transition can occur at zero temperature Sondhi et al. (1997). A quantum critical state of matter has been anticipated in the proximity of these transitions Sachdev (1999); Varma et al. (2002). This possibility has recently attracted much attention because the response of such a state of matter is expected to follow universal patterns defined by the quantum mechanical nature of the fluctuations Belitz et al. (2005). Candidates are for example found in heavy-fermion systems Coleman and Schofield (2005); v. Löhneysen et al. (2007) and high superconductors Varma et al. (1989). Quantum fluctuations play a dominating role in one-dimensional systems causing inter alia the breakdown of the Fermi-liquid into a Tomonaga-Luttinger (TL) liquid Giamarchi (2004a). Powerlaw behavior of the response functions is a natural consequence. Since , the phase Arg and are related by a Kramers-Kronig transformation. Due to the fact that is subject to the f-sum rule, we need for the integration to converge for . Since in addition is needed to have a convergent result for , the integral diverges for any value of . These divergencies can be avoided by limiting the powerlaw behavior to the range as in the expressionvan der Marel (1999)

(26)

The optical conductivity follows the relation , where the TL parameter characterizes the electron-electron interaction ( if the interaction is attractive), and is the order of commensurability ( at half filling and at quarter filling) Giamarchi (2008). This has been confirmed by experiments on the organic compound (TMTSF), where powerlaw behavior of the optical conductivity has been observed with , indicating a repulsive electron-electron interaction Schwartz et al. (1998). Recent pressure dependend studies of (TMTSF)AsF indicate a pressure dependence where increases from 0.13 (ambient pressure) to 0.19 (5 GPa), indicating a weakening of the electronic interaction Pashkin et al. (2006). A similar trend was reported by Lavagnini et al. (2009) for the CDW system LaTe, where the exponent in evolves from 1.6 to 1.3 when the pressure increases from 0.7 GPa to 6 GPa. Lee et al. (2005a) measured the optical conductivity for the chain contribution in YBaCuO, and observed a universal exponent in the doping range from .

No exact solutions are known up to date for interacting particles in two or three dimensions. However, the preponderance of quantum fluctuations diminishes as the number of dimensions is increased, and consequently the breakdown of the Fermi-liquid is not expected to be universal in dimensions higher than 1. An exception occurs when the system is tuned to a quantum phase transition. In this case a quantum critical state is approached, and powerlaw behavior of the optical conductivity

(27)

is a natural consequence for the response of charged bosons Fisher et al. (1990).141414When the spectral weight integral diverges for . The power law behavior behavior is therefore necessarily be limited to frequencies below some finite ultraviolet cutoff. Whether for fermions similar behavior should be expected is subject of intensive theoretical research Cubrovic et al. (2009). The limit of zero dissipation is described by . Experimentally Lee et al. (2002b); Kamal et al. (2006) was observed for the paramagnetic metal Cao et al. (1997) CaRuO while Kostic et al. (1998); Dodge et al. (2000a) for the 3D ferromagnet SrRuO with Curie temperature of K having a large magnetization of Ru Randall and Ward (1959); Callaghan et al. (1966); Longo et al. (1968) and the 3D helimagnetic metal MnSi Mena et al. (2003). Multiorbital correlations were shown to lead to an orbital non-FL metal with the observed frequency dependence Laad et al. (2008).

Figure 11: (Color online) (a) Temperature dependent of CaRuO shows the powerlaw scaling at three representative temperatures. The symbols in the inset mark the energy scale where powerlaw cease to hold. From Lee et al., 2002b. (b) Logarithmic plot of for SrRuO. The curves from the top correspond to temperatures K, 40, 60, and 80K, respectively. Dotted lines are fits to Eq. 27. From Dodge et al., 2000b.

At higher temperature, the powerlaw dependence of SrRuO is cutoff at a scale proportional to temperature, marked by dots in Fig. 11a. In contrast, in SrRuO the deviation from occurs at (see Eq. 27). At temperature higher than K, authors found a deviation from the formula (27) due to appearance of a downturn at low frequency. This gapping (not shown in Fig. 11b) might be connected with the similar low frequency downturn apparent also in CaRuO (Fig. 11a), which Lee et al., 2002b interpreted as a generic feature of the paramagnetic state of ruthenates.

In cuprate high-T superconductors one obtains, near optimal doping, the coefficient Schlesinger et al. (1990); El Azrak et al. (1994); van der Marel et al. (2003); Hwang et al. (2007). According to Eq. (27) the phase should be constant and equal to Baraduc et al. (1996); Anderson (1997). A crucial check therefore consists of a measurement of the phase angle of the optical conductivity, . A constant phase angle of 60 degrees up to at least 5000 cm is observed in optimally doped BiSrCaCuO van der Marel et al. (2003); Hwang et al. (2007), shown in Fig. 12.151515 The dielectric constant at finite frequencies is the superposition of the free carrier contribution, , which is the focus of this discussion, and ”bound charge” polarizability (see footnote 3 of section II.1), the onset of which is above 1.5 eV for the cuprates (see III.2. Using ellipsometry between 0.8 and 4 eV, and reflectance data between 0.01 and 0.8 eV van der Marel et al. (2003) obtained for optimally doped BiSrCaYCuO. Using reflectance spectra in a broad frequency range Hwang et al. (2007) obtained between 4.3 and 5.6 for BiSrCaCuO samples with different dopings.

Figure 12: (Color online) Universal power law of the optical conductivity and the phase angle spectra of optimally doped BiSr