Electrodynamics of correlated electron systems:Based on lectures first given at the 2008 Boulder School for Condensed Matter andMaterials Physics; revised for the 2018 QS3 Summer School @ Cornell University

N. P. ArmitageDepartment of Physics and Astronomy, The Johns Hopkins University, Baltimore, MD 21218, USA

(Dated: June 14, 2018)

Contents

I. Introduction 3

II. Formalism 5A. Optical Constants of Solids 5B. Classical Treatments: Drude-Lorentz 7C. The Quantum Case 11

III. Techniques 12A. Microwaves 12B. THz spectroscopy 14C. Infrared 16D. Visible and Ultraviolet 17

IV. Examples 19A. Simple Metals 19B. Semiconductors and Band Insulators 21C. Electron glasses 21D. Mott insulators 24E. Superconductors and other BCS-like states 25

V. Advanced Analysis and Techniques 28A. Sum Rules 28B. Extended Drude Model 29C. Frequency dependent scaling near quantum

critical points. 37D. Using light to probe broken symmetries 40E. Pump-probe spectroscopy and non-equilibrium

effects 41F. Topological materials and Berry’s phase 42

VI. Acknowledgments 45

References 46

Note added to 2018 update: These notes are based onlectures I first gave at the Boulder School for CondensedMatter Physics in 2008. They have been further updatedto be distributed at the QS3 summer school at CornellUniversity in the summer of 2018. I originally wrotethese detailed notes 1) for students at the Boulder schooland 2) for my own students and postdocs as I found my-self explaining some of the same material many times andI thought it would be useful to have some of my thoughtson these subjects written down. After the school’s con-clusion, I didn’t expect them to have much of a life be-yond my own research group although I did post a slightly

cleaned up version to the arXiv in 2009. I was then sur-prised and (very!) pleased to find that they’ve had sucha big impact. Many people (from beginning graduate stu-dents to senior faculty) have told me over the years howuseful they have found them as an introduction to theelectrodynamic response of materials with quantum coop-erative phenomena. This was very gratifying.

In preparation for 2018, it is quite interesting to lookback at what I thought was relevant and important for“correlated electron systems” in 2008 and what was is ab-sent. Some of this reflects my own prejudices and someshows how the field has moved on. In the original version,although I mention the quantum Hall effect, it was onlyin the context of the plateau transition being a model sys-tem for quantum phase transitions (QPT) and the con-sequences of dynamical scaling near a QPT. There is noreference to topological insulators or topology in the orig-inal version of these notes. Indeed the landmark paperof Hseih et al.1 had just been published and its signifi-cance was lost on me at the time. Needless to say thishas become a major topic of inquiry. And although mostexperimental aspects of the work on topological materi-als can be understood in terms of free fermions, intel-lectually the field has moved in tandem with the field ofstrong correlations and so I include some discussion onthese systems in the updated notes. Also, note that inthe 2008 version I left out all mention of nonlinear op-tics and nonequilbrium effects. There was little use of theformer at the time (with the exception of excellent workon magnetoelectrics and multiferroics2) and I felt that –despite some nice work3 – the latter was still too imma-ture to cover explicitly. That has changed for both thesetopics. Optics (both linear or nonlinear) has now madeimportant contributions in the detection of subtle brokensymmetry states of matter in cuprates and iridates4–7.There is also use of non-linear spectroscopies for 2D THz,which I consider tremendously promising9. And there hasbeen recent tremendous interest in non-equilbrium phaseswith – most notably – the possibility of driven nonequi-lbrium room temperature superconductivity in cupratesand K3C60

10,11. It is interesting to note that the interestin quantum criticality and scaling phenomena has fadedsomewhat although their importance is undiminished.

What is clear (and has not changed one whit) that isthat the field of optical effects of “quantum correlated sys-tems” continues to be tremendously active. The compat-ibility of optical probes with many different sample envi-ronments and the increased ease of use of ultrafast lasersystems means that there is a continuing vitality to the

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field. it is clear that optical probes of solids will continueto make essential contributions to the field for years tocome.

Physical and chemical systems can be characterizedby their natural frequency and energy scales. It is hardlyan exaggeration that most of what we know about suchsystems, from the acoustics of a violin to the energy lev-els of atoms, comes from their response to perturbationsat these natural frequencies. For instance, chemists andbiologists use infrared spectroscopy absorptions around1650 cm−1 (which corresponds to frequencies of 49 THz,wavelengths of 6 µm, and energies of 0.2 eV) to identifythe carbon-carbon double bond in organic compounds.And it was the observation of light emission from atomsat discrete energies in the electron Volt (eV) range thatled to the quantum theory.

It is of course the same situation in ‘correlated’ elec-tron materials. We can learn about the novel effects ofstrong electron-electron interactions and the propertiesof collective states of matter (superconductors, quantummagnets etc.) by characterizing their response to smallamplitude perturbations at their natural frequencies. Insolids, these natural frequency scales span an impres-sively large frequency range from x-ray down to DC. Thisincredibly broad range means that a blizzard of experi-mental techniques and analysis methods are required forthe characterization of correlated systems with opticaltechniques.

This short review and lecture notes attempt to lay outa brief summary of the formalism, techniques, and anal-ysis used for ‘optical’ spectroscopies of correlated elec-tron systems. They are idiosyncratic, occasionally opin-ionated, and - considering the breadth of the subject -incredibly brief. Unfortunately, there is no single com-plete treatise that presents a complete background forthese topics in the context of correlated electron materi-als. However, there are a number of excellent resourcesthat collectively give a solid background to this field.

I recommend:

1. F. Wooten, “Optical Properties of Solids”, (Aca-demic Press, New York, 1972).The truly excellent and classic introduction to thesubject of the electrodynamic response of solids (al-though there is very little in it in the way of elec-tronic correlations). The book that multiple genera-tions of spectroscopists learned from either directlyor indirectly. To say that it is full of only basicobvious things is akin to complaining that Mac-beth is full of quotations. Unfortunately it is outof print....12

2. M. Dressel and G. Gruner, “Electrodynamics ofSolids: Optical Properties of Electrons in Matter”,(Cambridge University Press, 2002).Much newer and modern with many plots of rele-vant response functions. Good discussion of manyimportant techniques13. Beware of typos.

3. G. D. Mahan, “Many-Particle Physics”, (Plenum,2nd ed., 1990).The go-to resource for perturbative treatments ofcorrelations in solids14. Excellent discussion ofsum rules.

4. Richard D. Mattuck, “A Guide to Feynman Dia-grams in the Many-Body Problem”, (Dover, 2nded., 1992).It has been called “Feynman diagrams for dum-mies.” Well ... I like it anyways15.

5. M. Tinkham, Introduction to Superconductivity,2nd Ed. , McGraw-Hill, NY, (1996).The classic exposition on superconductivity. Ex-cellent treatment of EM response of superconduc-tors (and other BCS-like states) both from the phe-nomenological two-fluid model and the full Matthis-Bardeen formalism16.

6. D. van der Marel, “Optical signatures of electroncorrelations in the cuprates”, “Strong interactionsin low dimensions, Series: Physics and Chem-istry of Materials with Low-Dimensional Struc-tures” Vol. 25, 237-276 (2004); Also available athttp://arxiv.org/abs/cond-mat/0301506Excellent treatment of underlying theory, particu-larly that associated with various sum rules. Manyrelevant examples18.

7. Andrew Millis, “Optical conductivity and corre-lated electron physics”, “Strong interactions in lowdimensions, Series: Physics and Chemistry of Ma-terials with Low-Dimensional Structures” Vol. 25,195-235 (2004); Also available at http://phys.columbia.edu/~millis/july20.pdfExcellent review covering many aspects of the the-ory of optical measurements for correlated electronphysics. Particularly good discussion on the use ofsum rules. Explicit calculations within Kubo for-malism. Many experimental examples with accom-panying theoretical discussion. Unique treatmenton the applicability of the conventional descriptionsof the coupling of EM radiation to solids19

8. D. Basov and T. Timusk, “Electrodynamics ofhigh-Tc superconductors”, Reviews of ModernPhysics 77, 721 (2005).A thorough review of the use of optical probes inthe cuprate superconductors20.

9. L. Degiorgi, “The electrodynamic response ofheavy-electron compounds”, Rev. Mod. Phys. 71,687 (1999).

10. A. Millis and P.A. Lee, “Large-orbital-expansionfor the lattice Anderson model”, Phys. Rev. B 35,3394 (1987).Experimental and theoretical papers with excellentsections on the phenomenological expectations forthe optical response of heavy fermion materials andoptical self-energies21,22.

11. C.C. Homes, “Fourier Transform InfraredSpectroscopy”, Lecture notes available herehttp://infrared.phy.bnl.gov/pdf/homes/fir.pdf

3

A good introduction to various technical issues as-sociated with Fourier Transform Infrared Reflectiv-ity (FTIR); the most commonly used measurementtechnique for optical spectra of correlated electronsystems23.

12. George B. Arfken and Hans J. Weber, “Mathe-matical Methods for Physicists” (Academic Press,2005).The classic textbook of mathematical methods forphysicists. Now with Weber. Good reference forKramers-Kronig and Hilbert transforms24.

13. John David Jackson, “Classical Electrodynamics”(3rd ed., Wiley, 1998).No motivation need be given25.

14. Dmitrii L Maslov and Andrey V Chubukov, “Op-tical response of correlated electron systems”, 2017Rep. Prog. Phys. 80 026503.Good treatment of many modern issues from a the-oretical perspective. Discussion of optical conduc-tivity of non-Fermi liquids27.

Other references as cited below.

I. INTRODUCTION

As mentioned above, the energy and frequency scalesrelevant for correlated systems span the impressive rangeof the x-ray down to DC (Fig. 1). For instance, atomicenergy scales of 0.5 eV to the keV make solids possi-ble through chemical bonding. These energies manifestthemselves explicitly in correlations by, for instance, set-ting the energy scale of the large on-site repulsions U (1- 10 eV) of electrons which can lead to Mott-Hubbardinteractions and insulating states. Typical overlap inte-grals between atomic wavefunctions in solids are at thelow end of the eV energy scale and set the scale of Fermienergies in metals and hence the energy scale for electrondelocalization and roughly that also of plasmon collectivemodes. Various other collective modes such as phonons(lattice vibrations) and magnons (magnetic vibrations)are found at lower energy scales, typically at fractionsof an eV. At energies of order 50 meV are the super-conducting gaps of optimally doped cuprates. At evenlower energies of the few meV scale the scattering rate ofcharges in clean metals is found. This is also the energyscales of gaps in conventional superconductors. Many lo-cal f -electron orbitals, which are relevant in Kondo ma-terials are found at these energies. At even lower scales(approximately 10’s of µeV) can be found the width ofthe ‘Drude’ peaks found in the AC conductivity and thesuperconducting gaps of clean heavy fermion systems.Finally the 1 - 10 µeV energy scale corresponds to thelowest temperatures typically reached with conventionalcryogenics in the solid state (10 - 100 mK). The corre-sponding frequencies become important when studyingcrossover phenomena near quantum critical points with~ω ≈ kBT . All such energy scales can be studied in a

number of different contexts with various photon spec-troscopies.

Although there are many different photon spectro-scopies that can be discussed that span these scales, inthese lectures I concentrate on ‘optical’ spectroscopies,which I define as techniques which involve transitionswith net momentum transfer q = 0 and whose absorptionand polarization properties are governed by the opticaldipole matrix element. I do not discuss the fascinat-ing and important work being done using other photonspectroscopies using light and charge in correlated sys-tems with for instance, Raman spectroscopy28, electronenergy loss, Brillouin scattering, optical Kerr rotation1,photoemission, and fluorescence spectroscopies to give atremendously incomplete list. I am also not going todiscuss the important work being done using THz lightto look at magnetic excitations (through the magneticdipole operator) in materials. The emphasis here is oncharge properties.

In the present case, the quantities of interest are typi-cally the complex frequency dependent conductivity σ(ω)or dielectric constant ε(ω). Alternatively data may beexpressed in terms of one of a variety of straightforwardcomplex parameterizations of these quantities like the in-dex of refraction n and absorption coefficient k, or thecomplex surface impedance Zs. These quantities are de-fined for the interaction of light with materials generi-cally from zero frequency to arbitrarily high frequencies.In these lecture notes, however, I confine myself to thefrequency range from microwaves, through the THz, toinfrared, visible and ultraviolet. These are the frequencyranges that correspond to various typical energy scalesof solids. Each of these regimes gives different kinds ofinformation and requires different techniques. For in-stance, microwaves are measured in cavities, striplines,or with Corbino techniques. THz is a huge growth areawith the advent of time-domain THz and the increaseduse of Backward Wave Oscillators (BWOs). The infraredand visible range are measured by Fourier Transform In-frared Reflectivity (FTIR). Visible and Ultraviolet canbe measured by grating spectrometers and spectroscopicellipsometry. Of course there are large overlaps betweenall these regimes and techniques.

I caution that throughout these lecture notes, I fre-quently make use of the language of quasi-free electronswith well-defined masses and scattering times. It is notclear that such a description should be automaticallyvalid in correlated systems. In fact, it is not even clearthat such a description should be valid in ‘normal’ ma-terials. Why should an ensemble of 1024 electrons/cm3

interacting with each other via long range Coulomb in-teraction have excitations and states that resemble any-thing like those of free electrons? Naively one would ex-pect that excitations would be manifestly many-electron

1 Kerr rotation is discussed below in the 2018 version

4

FIG. 1: (Color) The electromagnetic spectrum from radiofrequencies to gamma rays. Diagram from the LBL Advanced LightSource web site http://www.lbl.gov/MicroWorlds/ALSTool/EMSpec/EMSpec2.html.

composite objects wholly unrelated to the individual par-ticles which constitute the system. It is nothing short ofa miracle that, in fact, many materials are relatively welldescribed by the assumption that interactions do not playa principle role in the explicit physics. Low-energy exper-iments (e.g. DC resistivity, heat capacity, thermal con-ductivity etc.) on materials like sodium, gold, or siliconindicate that many aspects of such systems can be welldescribed by free-electron physics. The largest effect onelectrons appears to be the static field of the ionic coresand the average static effect of the other electrons. Inother materials, like for instance the heavy fermion com-pounds, experiments seem to indicate that the chargecarriers are free, but have masses many times larger thanthat of a bare electron. But even here at the lowest en-ergy scales there are no explicit signs that charger carri-ers are simultaneously interacting with 1023 other chargesand spins. In general, it is surprising that considering theclose proximity that many electrons have to each otherthat such interactions appear to be only quantitative, butnot qualitative perturbations on the free-electron physics.

In a system of interacting fermions, the relatively weakeffect that even potentially strong interactions have onthe underlying physics can be understood by realizingthe strong constraint that Fermi statistics and the exis-tence of a Fermi surface provides on the scattering kine-matics. This phase space constriction gives, within theconventional treatment, a scattering rate that goes like∼ ω2 + T 2 at low frequency and temperature. Due tothis quadratic dependence, for some small frequency theparticle’s scattering rate will be less than its energy andone can say that the quasi-particle excitation is well-

defined and scattering is a minor perturbation on thefree-electron physics. Such a perturbation renormalizesto zero in the ω → 0 limit. The effect of interactions canbe subsumed into giving quasiparticle excitations a finitelifetime and a renormalized energy parameterized by aneffective mass m∗.

Landau hypothesized that if one envisions slowly turn-ing on an interaction potential in a gas of non-interactingelectrons, some character of the original system wouldremain29. Specifically, he conjectured that there wouldbe a one-to-one correspondence between states and ex-citations of the non-interacting system and those of theinteracting system. In which case there can be said to bean adiabatic continuity between the two and one can tryto understand and model the interacting system by mod-eling the non-interacting one. A system with such a map-ping is termed a Fermi-liquid. Its quasi-free electron-like excitations are termed quasi− particles.

The success of Fermi liquid theory comes from, asmentioned above, the constraint on scattering kinemat-ics for low-energy excitations. This gives the result thatquasi-particle excitations are only defined at arbitrar-ily small energy scales. The condition for well definedquasiparticles to exists, βω2 ω, means that as one in-creases the parameter β, which characterizes the strengthof electron-electron excitations, the maximum energy ofwell-defined excitations decreases, but well-defined quasi-particles will still exist at low enough energies. Withinthis view, stronger interactions mean only that Fermi liq-uid behavior will occur at lower-energy scales i.e. at lowertemperatures and excitation energies.

Systems for which the Fermi-liquid paradigm is valid

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can be described at low energies and temperatures interms of quasi-free electrons. Although the vast ma-jority of normal metallic systems do seem to obey theFermi liquid phenomenology, it is unclear whether sucha description is valid for ‘correlated’ electron systems.A violation could occur for instance in Hubbard modelswhen the intersite hopping parameter t, which parame-terizes the band width, is much smaller than the onsiteCoulomb interaction U resulting in an insulating state.So far however the fractional quantum Hall effect is theonly case where Landau’s conjecture regarding a one-to-one correspondence of states has been experimentallyfalsified30,31. On the theoretical side, exact solutions to1D interacting models also show definitively that the fun-damental excitations in 1D are not electron-like at all,but are fractions of electrons: spinons and holons thatcarry spin and charge separately32. This would be an-other case where the electronic quasiparticle concept isnot valid, but thus far in real systems the residual higherdimensionality has been found to stabilize the Fermi liq-uid (See for instance Ref. 33)2

It is true that many systems that we call strongly cor-related exhibit a phenomenology inconsistent with theFermi-liquid paradigm. It is unclear whether this is be-cause interactions have driven such materials truly intoa non-Fermi liquid state or because the Fermi liquid phe-nomena is obscured at the experimentally accessible tem-peratures and frequencies. For instance, it is importantto keep in mind that the much heralded non-Fermi liquidbehavior of high Tc cuprates is actually exhibited at rela-tively high temperatures (∼100 K) above the occurrenceof superconductivity. It may be that the inopportune oc-currence of superconductivity obstructs the view of whatwould be the low-energy quasiparticle behavior.

At this point it is far from clear to what extent theFermi liquid paradigm is valid in many of the materialsbeing given as examples below. In this regard, one mustkeep in mind however, that despite the murky situationmuch of the language tossed around in this field presup-poses the validity of it and the existence of well-definedelectronic excitations. Indeed, much of the used terminol-ogy shows this bias. The terms ‘density of states’, ‘effec-tive mass’, ‘scattering rate’, ‘Pauli susceptibility’, ‘bandstructure’, ‘electron-phonon’ coupling all require the con-text of the Fermi liquid to even make sense. Clearly suchlanguage is inappropriate if such excitations and statesdo not exist! Despite this, in the literature one sees manypapers discussing, for instance, optical or photoemission

2 Carbon nanotubes, which are in principle perfect 1D systems,have been claimed to have tunneling conductances that obeya voltage - temperature scaling that is consistent with Lut-tinger (1D) liquid physics34. However there are serious questionswhether this study showed scaling to the requisite precision inorder to confirm a non-Fermi liquid nature. It may also be pos-sible to account for the data in terms of temperature dependentCoulomb blockade35.

electronic self-energies, while at the same time the au-thors discuss the non-Fermi liquid aspect of these mate-rials. It is not clear whether it is appropriate to discusselectronic self-energies in materials where the elementaryexcitations are not electron-like. It is the case that whilethe formalism for generating electronic self-energies fromoptical or photoemission data as discussed below maybe followed straightforwardly, the physical signficance ofsuch self-energies is not necessarily clear. For instance,although the parametrizations of optical spectra in termsof the frequency dependent mass and scattering rate fromthe extended Drude model (see below) can always bevalid as a parameterization, it is reasonable that onecan only assign physical significance to these quantitiesif the quasi-particle concept is valid in the energy rangeof interest.

Some aspects of the below formalism are model in-dependent and some rests on the concept of well-definedelectronic Fermi-liquid excitations. Although I will try tomake the various distinctions clear, I use the language ofquasi-free electrons almost entirely throughout the below,because at the very least it provides a rough intuition ofthe kinds of effects one expects in insulator, metals, andsuperconductors. It also provides a self-contained formal-ism for the analysis of optical spectra. A generalizationof these ideas to strongly correlated systems does not cur-rently exist. In the spirit of learning to walk before onelearns to run, I use the language of quasi-free electronsin these lecture notes throughout, but I caution on thenaive application of these ideas, which are only formallytrue for non-interacting systems to strongly interactingones! The generalization to non-Fermi liquid, correlated,strongly interacting etc. etc. systems is left as an excer-cise for the reader!!!

II. FORMALISM

A. Optical Constants of Solids

Any discussion of the interaction of light with matterstarts with Maxwell’s equations

∇ · E(r, t) = 4πρ(r, t), (1)

∇× E(r, t) = −1

c

∂

∂tB(r, t), (2)

∇ ·B(r, t) = 0, (3)

∇×B(r, t) =1

c

∂

∂tE(r, t) +

4π

cJ(r, t), (4)

where E and B represent the electric and magnetic fieldsaveraged over some suitable microscopic length (typicallythe incident light wavelength). As usual we introduceauxiliary fields as

D = E + 4πP, (5)

H = B − 4πM, (6)

6

where D and H have their conventional definitions andM and P are magnetization and polarization. Using thecontinuity equation for the electric current ∇ · J = −∂ρ∂tand accounting for all sources for free conduction, po-larization, and magnetization currents Jtot = Jcond +∂P∂t +c∇×M as well as the external and induced chargesρtotal = ρext+ρind we get the additional equations (pleaserefer to Jackson25 for a more extended discussion).

∇ ·D = 4πρext, (7)

∇×H =1

c

∂D

∂t+

4π

cJext +

4π

cJcond. (8)

Here we assume that we are in the linear regime andthe response of polarization, or magnetization of currentis linear in the applied field. We therefore write

P = χeE,

M = χmH,

J = σE. (9)

Typically we express the electric and magnetic suscep-tibilities in terms of dielectric functions ε = 1+4πχe andmagnetic permittivity µ = 1 + 4πχm. Except for explic-itly magnetic materials, χm = 0. The dielectric function(or its Fourier transform) is a response function that con-nects the field E at some time t and position r with thefield D at some later time and position. Generally wefirst define it in the time and position domain via therelation

D(r, t) =

∫ t

−∞

∫ε(r, r′, t, t′)E(r′, t′)d3r′dt′. (10)

One could also describe the system’s response in termsof current and conductivity σ(ω)

J(r, t) =

∫ t

−∞

∫σ(r, r′, t, t′)E(r′, t′)d3r′dt′. (11)

In this context, the real space and time dependent εand σ are usually referred to as memory functions for ob-vious reasons. Note that the principle of causality is im-plied here; effects cannot precede their causes. For anal-ysis of optical spectra we are typically more interestedin their Fourier transforms σ(q, ω) and ε(q, ω). This isbecause these are the quantities which are actually mea-sured, but also because the mathematical complicationsof convolution in Eqs. 10 and 11 is avoided; convolutionin the timed-domain becomes multiplication in the fre-quency domain. The conductivity and dielectric functionare related by3

3 One frequently sees the complex response function written usingthe real part of the conductivity and the real part of the dielectricfunction e.g. ε = ε1 + i4πσ1/ω.

σ =iω

4π(1− ε). (12)

Given the very general form of Eqs. 10 and 11, simplephysical considerations allow a number of general state-ments to be made. First, due to the vast mismatch be-tween the velocity of light and the typical velocity ofelectrons in solids, we are typically concerned with theq = 0 limit of their Fourier transforms4. This means thatexcept in a few circumstances, where one must take intoaccount non-local electrodynamics, (very pure metals orclean superconductors for example), one can assume thatdown to the scale of some microscopic length there is alocal relationship between the quantities given in Eqs. 9.This means that while these quantities may have spa-tial dependence (for instance the current J is confined tosurfaces in metals), the proportionality expressed in Eqs.9 holds. One can use the above expressions to rewriteMaxwell’s equations explicitly in terms of ε, µ and σ.(Please see Jackson25 for further details.)

Furthermore, the principle of causality - that effectscan not proceed their causes - demands strict temporalconsiderations regarding the integrals in Eqs. 10 and 11.This leads to the powerful Kramers-Kronig relations re-lating the real and imaginary part of such response func-tions. We can rewrite Eq. 11 as

J(t) =

∫ t

−∞σ(t− t′)E(t′)dt′. (13)

From causality the conductivity memory function σ(t−t′) has the property that σ(τ < 0) = 0 with τ = t − t′.The Fourier transform of σ(t− t′) is then

σ(ω) =

∫σ(τ)eiωτdτ. (14)

The integral can be performed in the complex fre-quency plane with the substitution ω → ω1 + iω2. Itis then written

σ(ω) =

∫σ(τ)eiω1τe−ω2τdτ. (15)

The factor e−ω2τ in this integral is bounded in theupper half of the complex plane for τ > 0 and in thelower half plane for τ < 0. Since σ(τ < 0) = 0, thismeans that σ(ω) is analytic for the upper half of thecomplex plane. This means that one can use Cauchy’stheorem and therefore the relation

4 Alternatively one may say it is the q = 0 limit is relevant be-cause of the large difference between the momentum carried byan ‘optical’ phonon and typical lattice momenta

7

∮σ(ω′)

ω′ − ω0dω′ = 0 (16)

holds in the usual way in the upper half of the complexplane. (Please see Wooten12, Jackson25 or Arfken24 forfurther details on the integration and derivation of theKramers-Kronig relations). One gets the expression

P

∫ ∞−∞

dω′σ(ω′)

ω′ − ω− iπσ(ω) = 0, (17)

where P as usual denotes the principal part.From this the Kramers-Kronig relations can be in-

ferred. Using the fact that in the time-domain Im[σ(τ)] = 0, which means that σ(−ω) = σ∗(ω) we canwrite a simpler form of them as

σ1 =2

πP

∫ ∞0

dω′ω′σ2(ω′)

ω′2 − ω2, (18)

σ2 = −2ω

πP

∫ ∞0

dω′σ1(ω′)

ω′2 − ω2. (19)

The Kramers-Kronig relations are tremendously use-ful in the analysis and determination of optical spectra.When one knows one component of a response functionfor all frequencies, the other component automaticallyfollows. As will be discussed below, they are used exten-sively in Fourier Transform Infrared Reflectivity (FTIR)measurements to determine the complex reflectivity (am-plitude and phase) when only measuring reflected power.

The above discussion was based on some generalizedfrequency dependent conductivity σ(ω) and its model in-dependent properties. In real materials, one has a zooof different possible contributions to the electromagneticresponse. For instance in simple metals with periodictranslational symmetry (i.e. a crystal), one can iden-tify a number of absorptions that satisfy the q = 0 con-straint. For the schematic band structure shown in Fig.2 (left), one expects a broad feature at finite energy (inred) in the optical conductivity (Fig. 2 (right)), whichcomes from the sum over all possible direct interbandband absorptions (red arrow) in which electrons are pro-moted from below EF across an energy gap to a higherlying band, with zero net momentum change. These areso-called vertical transitions. Near EF absorptions withlow but finite ω (green arrow) are only possible if stricttranslational symmetry has been broken by, for instance,disorder. Electrons moving in Bloch waves with meanfree path `, can violate strict momentum conservationin optical absorption at momentum scales on the orderof 2π/`. One can think of this heuristically as smear-ing out the band structure on this scale to allow verticaltransitions within the same band. This gives a peak cen-tered at zero frequency (in green). In the limit of perfecttranslational symmetry this peak would be a delta func-tion centered at ω = 0. One can also have excitations of

harmonic waves of the lattice (phonons) if such phononsposses a net dipole moment in the unit cell. They appearas distinct and frequently very sharp peaks in the opticalconductivity (in blue). Many different other excitationsexist as well that can potentially be created via opticalabsorption.

FIG. 2: (Color) Various different optical absorptions that sat-isfy the q = 0 constraint of optical spectroscopy can appear inthe optical conductivity of simple metals. Near EF intrabandabsorptions [green]. Interband absorptions [red]. Lattice vi-brations (phonons) [blue] .

Having introduced the general idea of optical responsefunctions, I now discuss the derivation of them, both clas-sically and quantum mechanically.

B. Classical Treatments: Drude-Lorentz

Almost the simplest model of charge conduction wecan conceive of is of a single charge e of mass m, drivenby an electric field E, and subject to a viscous dampingforce that relaxes momentum on a time scale τ . Considerthe force equation describing this situation

mx′′ = −eE −mx′/τ. (20)

If we assume harmonic motion then x = x0e−iωt and

E = E0e−iωt. Substituting in for x and E and solving

for x′0 we get

x′0 =eτE0

m

1

1− iωτ. (21)

If we then consider an ensemble of such charges withdensity N, and realize that the maximum current densityis J0 = Nex′0, we get the relation

J0 =Ne2τE0

m

1

1− iωτ. (22)

Using the previously defined relation J = σE we findthe relation for the frequency dependent ‘Drude’ conduc-tivity is

σ(ω) =Ne2τ

m

1

1− iωτ=Ne2τ

m

1 + iωτ

1 + ω2τ2. (23)

8

Interestingly, this classical model, first conceived of byPaul Drude, is actually of great use even in the analysisof particles obeying quantum mechanical statistics. Asdiscussed below one finds the same functional form toleading order in that case as well.

The Drude model (plotted in Fig. 3) demonstrates anumber of important features that are found generallyin response functions. Its limit at ω → 0 is Ne2τ/m iswell behaved mathematically and equivalent to the DCvalue. As shown in Fig. 3, it has real and imaginary com-ponents that differ from each other. At low frequencythe current’s response is in phase with the driving fieldand purely dissipative (real). At intermediate frequen-cies (when the driving frequency is of order the scatter-ing rate 1/τ), the real conductivity falls to half its DCvalue and imaginary conductivity peaks and equals thereal value. At higher frequencies, two things happen.Not only does the current begin to lag the driving field,but its overall magnitude decreases as the response hasa harder time keeping up with the driving electric field.The conductivity also exhibits distinct power laws in itsvarious frequency limits: linear for the imaginary part atlow ω, 1/ω at high ω, and 1/ω2 for the real part at highfrequency.

! = 1/15

! = 1/15

FIG. 3: (Color) Frequency dependent Drude conductivitywith scattering rate 1/τ = 15 on linear scale (top) log-logscale (bottom).

The Drude functional form can be used to capture thefinite frequency response of simple metals. See Fig. 4for measurements using time-domain THz spectroscopyof unintentionally doped n-type GaN. For high enoughdopings and at room temperature such simple semicon-ductors can be modeled as simple metals. One can seethe almost perfect agreement with the Drude form.

FIG. 4: Complex electric conductivity of unintentionallydoped n-type GaN. The dotted vertical line gives a scatteringrate of 0.81 THz. Lines are fits to the Drude expression ofEq. 23. One can see the almost perfect agreement with theDrude form. From Ref. 36.

In the limit of zero dissipation 1/τ → 0 Equation 23has the limiting form

σ1 =π

2

Ne2

mδ(ω = 0), (24)

σ2 =Ne2

mω. (25)

σ1 is zero everywhere, but at ω = 0. As σ1 is pro-portional to dissipation, this demonstrates that a gasof collisionless electrons cannot absorb photons at finitefrequency. This can be shown directly by demonstrat-ing that the Hamiltonian that describes this interaction,commutes with the momentum operator p and hence ithas no time dependence. With the exception of umklappscattering, electron-electron interactions do not changethis situation, as such interactions cannot degrade thetotal system momentum.

The dependence of the Drude conductivity on τ , N ,and m, shows that the optical mass of charge carriers inmetals can be obtained if for instance the charge densityis known from other parameters like the Hall coefficient.For instance, in Fig. 5 the mass determined by opticalconductivity is plotted against the linear coefficient in thespecific heat (which is proportional to the electronic den-sity of states and hence the mass). From simple metalsto exotic heavy fermion materials, it shows a dramaticlinear dependence over 3 orders of magnitude.

It is convenient to express the prefactors of the Drudeconductivity in terms of the so-called plasma frequency

ωp =√

4πNe2

m . The plasma frequency is equivalent to

9

FIG. 5: (Color) Specific-heat values γ vs effective massm∗, evaluated from the optical data using spectral-weightarguments37

the frequency of the free longitudinal oscillations of theelectron gas5. The conductivity can be expressed

σ(ω) =ω2p

4π

τ

1− iωτ. (26)

Therefore within the Drude model, the optical re-sponse is fully determined by two frequencies: the plasmafrequency ωp and the scattering rate 1/τ . Since ωp ismany orders of magnitude greater than 1/τ , this allowsus to define three distinctly different regimes for the op-tical response.

At low frequencies ω 1/τ , in the so-called Hagen−Rubens regime, the conductivity is almost purely real,frequency independent, and approximately equal to theDC value. At intermediate frequencies, in the relaxationregime where ω is on the order of 1/τ , one must explicitlytake into account the ωτ factor in the denominator ofEq. 23. As mentioned above when ω = 1/τ the realconductivity falls to half its DC value and the imaginaryconductivity peaks and is equal to the real value6. The

5 In actual materials, high energy excitations give a contribution tothe dielectric function ε∞ that renormalizes the observed plasma

frequency downward to the screened plasma frequency√

4πNe2

mε∞.

This is the actual frequency of plasmon oscillations in a realmaterial.

6 Of course in real materials, there are interband excitations whichcan also contribute to the conductivity. The Drude model aspresented here is an idealized case.

significance of the high frequency regime ω > ωp canbe seen in rewriting the conductivity as the dielectricfunction. It is

ε(ω) = 1−ω2p

ω2 − iω/τ. (27)

The real and imaginary parts of this expression aregiven in Fig. 6. One sees that for the conventional casewhere ωp 1/τ , the plasma frequency is the frequencyat which the real part of the dielectric function changessign from negative to positive i.e. ωp sets the scale forthe zero crossing of ε1. An analysis of the reflection andtransmission using the Fresnel equations (see Jackson25

), shows that above the zero crossing, metals describedby the Drude model become transparent. Hence this highfrequency limit is called the transparent regime.

Despite its classical nature, the Drude model describesthe gross features of many metals quite well at low fre-quencies. We give a number of examples of its use andextensions below. Of course, our interest in the electro-dynamics of solids extends far beyond the case of simplemetals. We will find the Drude model lacking in thisregard. And even for ‘simple’ metals, one has the inter-esting aspects of finite frequency absorptions that param-eterize band structure and, for instance, give copper andgold their beautiful colors. The Drude model is whollyinadequate to describe such finite frequency absorptions.For such processes in more complicated metals, semicon-ductors, and insulators, one can model them quantummechanically in a number of ways, but we can also gainpredictive power and intuition from an extension to theclassical Drude model called Drude-Lorentz.

FIG. 6: (Color) Frequency dependent Drude dielectric func-tion with scattering rate 1/τ = 15 and plasma frequencyωp = 60. The real part of the dielectric function changessign at ωp.

Here we envision the electrons are additionally alsosubject to a simple harmonic restoring force −Kx. Of

10

course, such a model is physically inadequate to describefinite frequency absorption in most band semiconductorsand insulators, as their insulating nature is due to theproperties of filled bands and not electron localization.Nevertheless, numerous aspects of absorption at ‘band-edges’ in semiconductors can be modeled phenomenolog-ically with Drude-Lorentz. Of course its applicability tothe modeling of harmonic phonon absorptions is obvious.

With ω20 = K/m, we extend Eq. 20 as

mx′′ = −eE −mx′/τ −Kx, (28)

Proceeding in exactly the same fashion as for the sim-ple Drude model, we obtain for the Drude-Lorentz con-ductivity

σ(ω) =Ne2

m

ω

i(ω20 − ω2) + ω/τ

, (29)

One can see that in the limit ω0 → 0 the Drude relationis obtained.

An important limiting case of the Drude-Lorentzmodel is obtained in the limit of large damping and largespring force. First consider Eq. 29 rewritten in terms ofthe plasma frequency and expressed as ε.

ε(ω) = 1 +ω2p

ω20

1

1− iωτω2

0− ω2

ω20

, (30)

Here we have factored the resonance frequency ω0 outof the denominator. Note that Eq. 30 written in thisfashion expresses the fact that the denominator of thedielectric function can be written as an expansion in pow-ers of −iω. By inspecting the frequency dependence ofa dielectric response, one can determine what terms aredominating in the classical equations of motion. Takingthe limit of this expression where ω2

0 →∞ and 1/τ →∞but in such a fashion that quantity τω2

0 stays finite, onehas the expression,

ε(ω) = 1 +ω2p

ω20

1

1− iωτω2

0

. (31)

This expression defines a new frequency scale Γ = τω20

that we can write as

ε(ω) = 1 +Ne2

4πK

1

1− iω/Γ. (32)

Eq. 32 is equivalent to the expression of Debye re-laxation, which is a common form for polarization re-laxation phenomena in insulators, particularly glasses.It describes the situation where a particle subject to astrong restoring force moves through a medium that is

so viscous so as to longer have inertial forces be rele-vant. In this regard, note that in the form given above,the particle mass has dropped out of the expression. Anuncommonly broad “resonance” peak is produced. Anequivalent form to this expression is found commonly formagnetic dipole relaxation as well. Also note that al-though Eq. 32 can be found to match large classes ofexperimental data quite well, it is not strictly speakinga fully mathematically consistent response function. Forinstance, it doesn’t satisfy the sum rules discussed be-low (Sec. V A) due to the neglect of the ω2/ω2

0 termin the denominator of Eq. 30. Typically the Debyemodel is derived for relaxing electric dipoles in a timedependent electric field, with a neglect of their momentof inertia38. The above derivation is different, but resultsin the equivalent mathematical form of Eq. 32. Exten-sion of the model exist39 for situations where inertial ef-fects are important (high frequencies and for fulfillmentof the sum rules). In recent work we found that sucheffects were important in the analysis of optical data forquantum spin-ice materials and were interpreted as beingconsistent with inertial effects of the effective magneticmonopoles40,41 in those systems.

! = 1/15 "0 = '5

FIG. 7: (Color) Frequency dependent Drude-Lorentz complexconductivity with scattering rate 1/τ = 15 and oscillator res-onant frequency ω0 = 45. The real conductivity is zero atω = 0 and positive for all frequencies.

As mentioned above, despite its classical nature andthe inapplicability of the underlying physical picture forsome situations, in some cases the Drude-Lorentz modelcan be used to quantify finite frequency absorptions. Inthe case of the band edge absorption in semiconduc-tors, a number of oscillators of the form of Eq. 29with different weights can be used to capture the de-cidely non-Lorentzian line shape. One can convince one-self that since any lineshape can be fit arbitrarily wellusing an arbitrarily large number of oscillators and more-over since the imaginary part of the response follows froma Kramers-Kroning transform of the real part, it is al-ways perfectly feasible to parametrize the response using

11

Drude-Lorentz.

C. The Quantum Case

The above Drude and Drude-Lorentz models are clas-sical models used to describe intrinsically quantum me-chanical phenomena and the fact that they are usefulat all to describe solids is surprising. It is clear thata quantum mechanical treatment is desired. The mostcommonly used method for the quantum mechanical cal-culation of electromagnetic response of materials is theKubo formalism14,42,43. It is based on the fluctuation-dissipation theorem, which relates the spontaneous fluc-tuations of a system described via its correlation func-tions to its driven linear response. The derivation hereloosely follows the one of Wooten12, which is mirrored inDressel and Gruner13.

We calculate conductivity, by calculating the absorp-tion rate of a system based on the power dissipatedP = σ1E

2. We first use Fermi’s Golden rule to calculatethe scattering probablility that incident radiation excitesan electron from one state |s〉 to another |s′〉

Ws→s′ =2π

~2|〈s′|Hi|s〉|2δ(ω − ωs′ + ωs). (33)

|s〉 and |s′〉 are the system’s eigenstates, which mayinclude the effects of many body physics. The interactionHamiltonian for charge with the electromagnetic field tofirst-order in field is

Hi =e

2mc

N∑i=1

[pi ·A(ri) +A(ri) · pi]− eN∑i=1

Φ(ri), (34)

where p, A, and Φ are the quantum mechanical opera-tors for momentum, vector potential, and scalar poten-tial and c is the speed of light. Since the vector potentialA depends on position, it does not in general commutewith p. However since p = −i~∇, one can show thatp · A − A · p = −i~∇ · A and since within the Coulombgauge ∇ · A = 0, p and A can in fact commute. Thismeans that for purely transverse waves with Φ = 0 theinteraction Hamiltonian simplifies to

HTi =

e

mc

N∑i=1

pi ·A(ri). (35)

By substituting in for the canonical momentum p =mv−eA/c, dropping all terms that are higher than linearin A, and replacing the summations by an integral oneHi takes the form

HTi = −1

c

∫drJT (r) ·AT (r). (36)

This gives a Fourier transformed quantity of

HTi = −1

cJT (q) ·AT (q). (37)

This means we can substitute in for the form of Hi inthe expression for Ws→s′ to get

Ws→s′ =2π

~2|〈s′|HT

i |s〉|2δ(ω − ωs′ + ωs). (38)

The matrix element in this expression follows from theform of HT

i . It is

〈s′|HTi |s〉 = −1

c〈s′|JT (q)|s〉AT (q). (39)

which leads to

Ws→s′ =2π

~2c2〈s′|JT (q)|s〉〈s|JT∗(q)|s′〉, (40)

|AT (q)|2δ(ω − ωs′ + ωs).

where JT∗(q) = JT (−q). One then sums over all occu-pied initial and all empty final states W =

∑s,s′ Ws→s′ .

The dissipated power per unit time and volume at a par-ticular photon frequency ω follows after a few mathemat-ical steps as

P = ~ωW = (41)

|AT (q)|2∑s

ω

~c2

∫dt〈s|JT (q, 0)JT∗(q, t)|s〉e−iωt.

The electric field is related to the vector potential asET = iωAT /c for transverse waves. Upon substitutionwe get the absorbed power per unit volume per unit timeexpressed as a current-current correlation function

P = |ET (q)|2∑s

1

~ω

∫dt〈s|JT (q, 0)JT∗(q, t)|s〉e−iωt.(42)

Using our previously given relation P = σ1E2, the

expression for the real part of the conductivity is

σT1 (q, ω) =∑s

1

~ω

∫dt〈s|JT (q, 0)JT∗(q, t)|s〉e−iωt.(43)

The imaginary part of σ(ω) follows from the Kramers-Kronig relation. This is the Kubo formula. It has theform expected for the fluctuation-dissipation theorem,that of a correlation function (here current-current) av-eraged over all the states |s〉 and describes fluctuationsof the current in the ground state. The conductivity de-pends on the time correlation between current operatorsintegrated over all times.

12

The above formalism is general and applies to any setof system states |s〉. An extension can be made in caseswhere Fermi statistics applies. The end result is (seeDressel and Gruner13 for the full derivation)

σT1 (ω) =πe2

m2ω

2

(2π)3|〈s′|p|s〉|2Ds′s(~ω). (44)

Here 〈s′|p|s〉 is known as the dipole matrix element.Ds′s is the so-called joint density of states, defined as

2(2π)3

∫δ(~ω−~ωs′s)dk. This equation, which is often re-

ferred to as the Kubo - Greenwood formula, is extremelyuseful for the calculation of higher lying interband tran-sitions in metals and semiconductors when the states |s〉and |s′〉 belong to different bands (as for instance illus-trated by the red arrow in Fig. 2). It also has the formthat one may naively expect; it is proportional to thejoint density of states and a matrix element which incor-porates aspects like symmetry constraints and selectionrules for allowed transitions.

We can use the Kubo formalism (Eq. 43) to calculatethe leading order metallic conductivity in a slightly morerigorous fashion than was done in the classical approachabove. We start with Eq. 43 and then posit as an ansatzthat finite currents relax as

J(q, t) = J(q, 0)e−t/τ . (45)

and assume that the current correlation time τ has nodependence on q. Inserting this into the Kubo formula,one gets that

σ1(q, ω) =1

~ω∑s

∫dte−iωt−|t|/τ 〈s|J2(q)|s〉. (46)

One now needs an expression for the current fluctu-ations at finite q. The expectation value for J is zero,but fluctuations lead to a finite value of 〈J2〉. Forsmall q, one can show in the ‘dipole approximation’ thatJ(q) = − e

m

∑i pi where i labels the individual charges.

With the reinsertion of a complete set of states s′ onegets the expression

σ1(q, ω) =e2

m2~ω

∫dte−iωt−|t|/τ

∑s,s′,i

|〈s′|pi|s〉|2. (47)

The quantity 2∑s,s′,i |

|〈s′|pi|s〉|2m~ωs,s′

found in Eq. 47 is

called the oscillator strength fs,s′ . Here ω in Eq. 47is the energy difference between states |s〉 and |s′〉 as isωs,s′ in the definition of the oscillator strength. Below,we pay special attention to the oscillator strength whenwe calculate the optical sum rules.

The oscillator strength can be easily calculated for the

case of free electrons where the energy is ~ω = ~2k2

2m and

the average momentum is |〈s′|pi|s〉|2 = ~2k2

4 . In such a

case fs,s′ = N the total number of free charge carriers.After performing the integral in Eq. 47 the final result forthe leading order conductivity calculated via the Kuboformalism is

σKubo1 (ω) =Ne2τ

m

1

1− iωτ, (48)

which is the exact same result as calculated classicallyvia the Drude model above. For more details see thechapters in Wooten12 and Mahan14.

III. TECHNIQUES

As mentioned above, the incredibly large spectralrange spanned by the typical energy scales of solids meansthat many different techniques must be used. Below Idiscuss the most used techniques of microwave measure-ments, THz spectroscopies, optical reflectivity, and ellip-sometry that are used to span almost six orders of mag-nitude in measurement frequency. These various regimesare loosely defined below, but of course there are overlapsbetween them7.

A. Microwaves

A standard measurement tool in the microwave range(100 MHz - 100 GHz) is microwave resonant cavity tech-niques. At lower frequencies (kHz and MHz), measure-ments can be performed by attaching contacts to samplesand the complex conductivity can be measured by lock-in techniques, network analyzers, and impedance analyz-ers. Such methods become problematic in the GHz rangehowever, as wavelengths become comparable to variousmeasurement dimensions (sample size, microwave con-nector dimensions, cable lengths), and capacitive and in-ductive effects become appreciable. In microwave cavityresonance techniques, one is less sensitive to such consid-erations as the sample forms part of a resonance circuit,which dominates the measurement configuration. Typi-cally one measures transmission through a cavity, whichis only possible when at resonance. The technique iswidely used for the study of dielectric and magnetic prop-erties of materials in the GHz range, because of its highsensitivity and relative simplicity.

7 Although I don’t discuss them, one might naturally include vari-ous radio frequency techniques in this list (which would increasethe frequency range by another three orders of magnitude atleast at the low end). At typical experimental temperatures,the low frequency available in techniques like mutual inductancemeasurements allows almost the DC limit of quantities like thesuperfluid density in superconductors to be measured. Interestedreaders are directed to Ref. 44 and reference therein.

13

A microwave resonator is typically an enclosed hol-low space of cylindrical or rectangular shape machinedfrom high-conductivity metal (typically copper or su-perconducting metals) with interior dimensions compa-rable to the free-space wavelength. On resonance, anelectromagnetic standing wave pattern is set up insidethe cavity. One characterizes the resonance characteris-tics of the cavity, both with and without a sample in-side. Upon sample introduction the resonance’s centerfrequency shifts and it broadens. From the shifting andbroadening, one can define a complex frequency shift,which with knowledge of the sample size and shape al-lows one to quantify the complex conductivity directly.For instance, in the limit of very thin films, the centerfrequency shift is proportional to σ2 and the broaden-ing is proportional to σ1. For samples much thicker thanthe skin or penetration depth, different relations apply45.The fact that the measurements are performed “on res-onance” means that a very high sensitivity is achieved.The technique does have the considerable disadvantagethat only discrete frequencies can be accesses, as one islimited to the standing wave resonance frequencies of thecavity, which means that true spectroscopy is not possi-ble. In such measurements one typically determines asample’s complex conductivity as a function of tempera-ture at some finite fixed frequency (See Figs. 32, 33 and34).

In contrast, the Corbino geometry is a measurementconfiguration that is capable of broadband microwavespectroscopy. It is compatible with low T and high fieldcryogenic environments and capable of broadband mi-crowave spectroscopy measurements from 10 MHz to 40GHz. It is a relatively new technique, but one whichhas recently been used with great success by a number ofgroups in the correlated electron physics community46–51.Although powerful, it has the disadvantage that its useis confined to samples which have typical 2D resistancewithin a few orders of magnitude of 50 Ohms.

In a Corbino geometry spectrometer a microwave sig-nal from a network analyzer is fed into a coaxial trans-mission line. A schematic is shown in Fig. 8 (left). Thesignal propagates down the coaxial line and is reflectedfrom a sample that terminates an open ended coaxialconnector. The network analyzer determines the com-plex reflection coefficient S11 of the sample which can berelated to the complex sample impedance ZL via the rela-tion S11 = ZL−Z0

ZL+Z0where Z0 is the coaxial line impedance

(nominally 50 Ω). Because the sample geometry is welldefined, knowledge of the sample impedance yields in-trinsic quantities like the complex conductivity. In thetypical case of a thin film where the skin or penetrationdepth is much larger than the sample thickness d, thesample impedance is related in a straightforward fashion

to the complex conductivity as σ = ln (r2/r1)2πdZL

where r1

and r2 are the inner and outer conductor radii respec-tively.

An experimental challenge in Corbino geometry mea-surements is that the coaxial cables and other parts of

FIG. 8: (Color) (left) Experimental geometry for a Corbinospectrometer. (right) The real and imaginary parts σ1+iσ2 ofthe optical conductivity spectrum of UPd2Al3 at temperatureT = 2.75 K. The data are fit to σdc(1 − iωτ)−1 with σdc =1.05×105(Ω−cm)−1 and τ = 4.8×10−11s and show excellentagreement with the Drude prediction. Adapted from Refs46,47

the transmission lines can have strongly temperature de-pendent transmission characteristics. Errors in the in-trinsic reflection coefficient, coming from standing wavereflections or phase shifts and damping in the transmis-sion lines, are accounted for by performing a number ofcalibration measurements. It is imperative that the samecryogenics conditions are reproduced between calibrationand each subsequent measurement. In general this is eas-ily done by ensuring the same starting conditions andusing a computer controlled cool-down cycle. Recentlyit has been demonstrated that a three sample calibra-tion (open, short, and a standard 50 ohm resistor) at alltemperatures dramatically increases the precision of thetechnique over previous single sample calibrations46.

As an example of the power of the technique we showin Fig. 8 (right) recent measurements taken on films ofthe heavy-fermion compound UPd2Al3 at T = 2.75 K.The data show almost ideal Drude behavior with a realconductivity of Lorentzian shape and an imaginary con-ductivity that peaks at the same ω as the real conductiv-ity has decreased by half. The anomalous aspect is theDrude peak is seen to be of remarkably narrow width -almost 500 times narrower than that of a good metal likecopper. This is a consequence of the mass renormaliza-tions in the heavy fermion compounds and a feature thatwould have been completely undetectable with a conven-tional spectrometer.

Recently there also been the very interesting develop-ment of a bolometric technique for high-resolution broad-band microwave measurements of ultra-low-loss samples,like superconductors and good metals52. This techniqueis a non-resonant one where the sample itself is actuallyused as a detector in a bolometric fashion. Microwavesare fed through a rectangular coaxial transmission lineinto which the sample under test and a reference sampleare mounted. Small changes in the sample and reference’stemperature are monitored as they absorb microwave ra-diation. The key to the success of this technique is thein situ use of this normal metal reference sample whichcalibrates the absolute microwave incident power. As the

14

absorbed power is proportional to the surface resistanceRs(ω), the independent measure of the incident powerallows one to measure the surface resistance precisely.

In such an apparatus, the sample temperature can becontrolled independently of the 1.2 K liquid-helium bath,allowing for measurements of the temperature evolutionof the absorption. The minimum detectable power ofthis method at 1.3 K is 1.5 pW, which corresponds toa surface impedance sensitivity of ≈ 1µΩ for a typical1mm× 1mm platelet sample. The technique allows verysensitive measurements of the microwave surface resis-tance over a continuous frequency range on highly con-ducting samples. This is a region of sample impedancesgenerally inaccessible with the Corbino technique andonly available at discrete frequencies in microwave cavi-ties.

A disadvantage of the technique is that although onemeasures Rs(ω) the quantity of interest is typically thecomplex conductivity σ(ω). One must make variousassumptions to get this from the relation for surfaceimpedance

Zs = Rs + iXs =

√iωµ0

σ1 + iσ2. (49)

as the surface reactance Xs is not measured. At low tem-peratures and frequencies in the superconducting state σ2

is mostly determined by the superconducting responseand can be related to the independently measured pen-etration depth λ. σ1 then follows from the relationRs = 1

2µ20ω

2λ3σ1. At higher temperatures and frequen-cies a more complicated iterative procedure must be usedthat is based on the Kramers-Kronig transform52.

This technique has been used to measure the real partof the conductivity of YBa2Cu3O6.5 single crystals acrossa wide frequency range as shown in Fig. 9. One can see aquasi-Drude like peak at low frequencies that comes fromquasi-particle excitation at the nodes of the d-wave su-perconductor. The peak has a narrow width of the orderof 5 GHz, signifying a collapse of quasiparticle scatteringin the superconducting state and very long quasiparti-cle scattering lengths. This is indicative of the very highquality of these YBa2Cu3O6.5 single crystals. Measure-ments of a broadband nature on these kind of materialsare not possible with other techniques. As powerful asit is, the technique is hampered by the fact that variousassumption must be made about σ2. This is problematicfor materials where its dependence is not known a priori.

In addition to the ones detailed above, there aremany other techniques which have been used for ma-terials characterization in the microwave range. Theseinclude microstrip and stripline resonators, transmissionline wave bridges, radio frequency bridges, parallel-plateresonators, and helical resonators. See Ref. 13,53,154 forfurther details.

FIG. 9: The real part of the microwave conductivity σ1(ω)extracted from measurements of Rs(ω) as described in thetext and Ref. 52. Figure adapted from52.

B. THz spectroscopy

Time scales in the picosecond (10−12 sec) range areamong the most ubiquitous in condensed matter systems.For example, the resonant period of electrons in semicon-ductors and their nanostructures, the scattering times ofelectrons in metals, vibrational frequencies of molecularcrystals, superconducting gap energies, the lifetime of bi-ologically important collective vibrations of proteins, and- now - even the transit time for an electron in Intel’snew THz transistor - these are all picosecond phenom-ena. This ubiquity means that experimental probes em-ploying Terahertz (THz) electromagnetic radiation arepotentially quite powerful. It is unfortunate then thatthis spectral range lies in the so-called ‘Terahertz Gap’- above the capabilities of traditional electronics and themicrowave range, but below that of typical optical in-strumentation. The THz spectral region has been a tra-ditionally difficult part of the electromagnetic spectrumto work in because of a number of reasons including weaksources, long wavelengths, and contamination by ambientroom temperature black body radiation. In recent years,however, there have been a number of developments thatallow measurements in the THz range in a manner thatwas not previously possible. These come in the form oftime domain THz spectroscopy and the increasing preva-lence of Backward Wave Oscillator (BWO) based spec-trometers. In recent years, however, a number of dra-matic technical advances such as time-domain THz spec-troscopy (TDTS) using so-called ‘Auston’ switch genera-tors and detectors have enabled measurements that spanthis gap in measurement possibilities.

THz spectroscopy has become a tremendous growth

15

field54, finding use in a multitude of areas including char-acterization for novel solid-state materials55,56, optimiza-tion of the electromagnetic response of new coatings57,probes of superconductor properties58,59, security ap-plications for explosives and biohazard detection60, de-tection of protein conformational changes61, and non-invasive structural and medical imaging62–64.

FIG. 10: A diagram of a typical experimental layout used inTDTS.

FIG. 11: An optical image of the two-contact photoconductive‘Auston’ switch antenna structure used in TDTS. Two elec-trodes are grown on top of an insulating high defect semicon-ductor, like low temperature grown GaAs or radiation dam-aged silicon. Electrode spacing is typically on the order of 10microns.

TDTS works (See Ref. 65 for an additional excellentshort summary) by the excitation of a source and acti-vation of a detector by ultrafast femtosecond laser (typ-ically Ti-sapphire) pulses. A basic schematic is shownin Fig. 10. A femtosecond laser has, via a beam split-ter, its radiation split off to fall on source and detec-tor Auston switches, which are typically pieces of lowtemperature-GaAs or radiation damaged silicon with twoelectrodes grown on top in a dipole arrangement andseparated by approximately 20 µm (Fig. 11). Beforelaser illumination, the switch has a resistance of a fewmegaohms. After the fast illumination by the femtosec-ond laser pulse, the source switch’s resistance falls to afew hundred ohms and with a bias by a few tens of volts,charge carriers are accelerated across the gap on a time

FIG. 12: (Color) (left) Time-domain trace of electric fieldtransmission through 60 µm thick SrTiO3 subtrate. Multiplereflections from front and back surfaces are readily visible.(right) Power spectrum of the transmission through the same60 µm thick SrTiO3 subtrate. The data is obtained from thesquared Fourier transform of the data in (left) and ratioed toa reference aperature signal. From Ref. 66

scale of a few picoseconds. Their acceleration produces apulse of almost single-cycle radiation, which then propa-gates through free space and - collimated by mirrors andlenses - interacts with the sample. Measurements havebeen typically performed in transmission. After passingthrough the sample, THz radiation falls on the secondAuston switch, which is also activated by the femtosec-ond pulse. Whereas the first Auston switch was DC bi-ased, the 2nd Auston switch is biased by the electric fieldof the THz radiation falling upon it. Current can flowonly across the second switch with a direction and mag-nitude proportional to the transient electric field at theinstant the short femtosecond pulse impinges on it. Theexperimental signal is proportional to this current witha magnitude and polarity that reflects the THz electricfield at the switch. A delay line is then advanced andthe THz electric field at a different relative times can bemeasured. In this way, the entire electric field profile asa function of time can be mapped out as shown in Fig.12(left). The time-domain pulse is then Fourier trans-formed to get the complex electric field as a function offrequency. A reference measurement is also performedon an aperture with no sample. The measured transmis-sion function is the ratio of the signal transmission to thereference [Fig. 12(right)].

There are a number of unique aspects to TDTS thatallow it to work exceptionally. Since the detected signalis proportional to the instantaneous electric field, andnot the power, the measured transmission function is thecomplex transmission coefficient for the electric field,which has both an amplitude and a phase. This allowsone to invert the data directly to get the real and imagi-nary optical constants (e.g. the complex conductivity) ofthe material. This is essential for a thorough characteri-zation of material’s properties. Typical optical measure-ments discussed below measure only reflected or trans-mitted power, quantities in which the complex opticalconstants of interest are mixed into in a non trivial fash-ion. In such measurements one has to measure over thelargest frequency range possible (even if one is only inter-

16

ested in a limited spectral range), extrapolate to both DCand infinite frequency, and then Kramers-Kronig trans-form to get phase information. In TDTS one can getboth components by direct inversion.

TDTS is also capable of unprecedentedly high signal-to-noise ratios. Typically efforts in the THz and far in-frared spectral range are complicated by a very largeambient black body radiation background. In the caseof TDTS the fact that the signal is ‘time-gated’ and co-herent, whereas black-body radiation is incoherent, al-lows a very high detection efficiency. Additionally, sincethe detected quantity is actually electric field and notpower (proportional to electric field squared), the noisein the spectral power is reduced by a factor proportionalto the electric field itself. These aspects allow detectionof transmission signals approaching one part in 106. Thisis an incredibly large dynamic range and is essentially un-precedented in optically based spectroscopies where onepart in 103 is typically considered extremely good.

TDTS was used for instance in the measurement ofthin Bi2212 cuprate superconducting films to show ev-idence for phase fluctuations above Tc. Corson et al.measured transmission through thin films and using theunique phase sensitivity of TDTS could show that therewas an enhanced σ2 (Fig. 13) even at temperatures wellabove Tc

59. This was interpreted as a finite phase stiff-ness on short length and time scales. Superconductingfluctuations as such should manifest themselves as a fi-nite superfluid stiffness measured at finite frequency. Theimaginary part of the optical conductivity is sensitive tosuperfluid stiffness through the relation σ2 = σQ

kBTθ~ω ,

where σQ = 4e2

hd is the quantum of conductance forCooper pairs divided by a length scale (typically the c-axis lattice constant) and Tθ is the generalized frequencydependent superfluid stiffness. Here, the superfluid stiff-ness Tθ is an energy scale expressed in temperature units.The stiffness is the energy scale to introduce phase twistsin the superfluid order parameter Ψ = ∆eiφ.

The other major development for THz spectroscopyis the increased prominence of Backward Wave Oscil-lators (BWOs) Although developed in the 60’s, BWOsare gaining increasing prominence in the investigation ofcorrelated systems in the important THz range67. TheseBWOs are traveling wave guide tubes, capable of pro-ducing very monochromatic THz range radiation over arelatively broad range per device. With a number oftubes, it is powerful method to cover the 0.03 THz to 1.5THz range in a continuous wave configuration.

The longer wavelengths in this spectral region give thecapabilities to measure phase of transmitted waveformsthrough Mach-Zender two-beam polarization interferom-etry. This method also allows direct precision measure-ments of both real and imaginary components of the com-plex optical response without resort to Kramers-Kronigtransforms and their associated ambiguities. A spectrom-eter based on BWOs is easily integrated with a low tem-perature cryostat and magnetic field system as well asglove boxes for environmental control.

FIG. 13: The dynamic (frequency dependent) phase stiffnessTθ(ω) as a function of temperature T. Data are shown fortwo samples, one with Tc 33 K. The data for each samplecorrespond to measurement frequencies of 100, 200 and 600GHz. Each set of curves identify a crossover from frequency-independent to frequency-dependent phase stiffness. Thedashed line shows that the crossover is set by the Kosterlitz-Thouless-Berenzkii condition for two-dimensional melting.From Ref. 59

C. Infrared

Fourier transform infrared reflectivity (FTIR) is theworkhorse spectroscopy for optical characterization ofsolids (Fig. 15). Measurements are possible from ∼ 10cm−1 to approximately 25,000 cm−1, although measure-ment get increasingly difficult below 50 cm−1. Heroic ef-forts can push the lower end of this range slightly below10 cm−168. Here the quantity of interest is typically thepower reflectivity (transmission measurements are possi-ble as well). One shines broadband light, sometimes fromseveral different sources, on a sample surface and via aninterference technique measures the reflected intensity.As the source spectrum and detector response can haveall kinds of frequency structure, it is necessary to ref-erence the reflected signal against a standard sample toobtain the absolute reflectivity. Typically the referencespectrum is chosen to be a noble metal like gold whosereflectance can to good approximation be taken to beunity over a broad frequency region below the material’splasma frequency (see below). Typically gold is evapo-rated onto a sample as shown in Fig. 14 or the sample isreplaced with a mirror to perform this referencing.

As the materials we are interested in typically haveinteresting experimental signatures in both the real andimaginary response, it is not sufficient to to compare totheory by simply measuring quantities like absorbence.We want quantities like the complex conductivity or di-electric function. It is not necessarily straightforwardto obtain a complex quantity from the measured scalar

17

FIG. 14: (Color) Absolute reflectivity R(ω) in FTIR is ob-tained by measuring the reflected intensity and ratioing withthe intensity after gold evaporation or from a reference mirror.Figure courtesy of A. Kuzmenko.

magnitude of the reflectivity R(ω). Unlike TDTS, phaseis not measured in FTIR measurements. Therefore typi-cally use is made of the above discussed Kramers-Kronigtransforms, which apply to any causal response function.If one knows one component of the response (real or imag-inary) for all frequencies then one can determine theother component. For reflected power, one can obtainthe reflected phase as

φ(ω) = − 1

πP

∫ ∞−∞

dωln|R(ω)|ω′ − ω

. (50)

Of course the problem is that one doesn’t measureover an infinite frequency range. In typical FTIR spec-troscopy, the usual mode of operation is therefore to mea-sure over as large a frequency range as possible and thenextrapolate with various schemes to ω → 0 and ω →∞.This method works quite well for some materials, al-though it can generate large errors when, for instance,reflectivities approach unity as they do in good conduc-tors at low frequencies8. It can also be rather inefficientand cumbersome as one must measure out to the multi-eV range even if one is interested in meV level phenom-ena.

As mentioned, FTIR spectroscopy is a standard spec-troscopic technique for the characterization of materi-als and chemicals. It is used routinely to identify thepresence of various chemical bonds in chemistry as in,for instance, C = C or C = O, which all have dis-tinct frequencies and absorptive strengths. It is also thestandard optical tool for probing materials like high-Tcsuperconductors20. It has the advantage of being rela-

8 See Ref. 69 for a good comparison between FTIR and time-domain THz data at low frequencies on a good metal.

tively easy to perform and possessing a very large spec-tra range. It has the above mentioned disadvantage ofonly measuring a scalar - the power - and having a lowerbound on the spectral range that is at the limit of thatexplicitly relevant for many correlated materials.

FIG. 15: (Color) Modified Fourier Transform SpectrometerBruker 66. As shown, the spectrometer is setup to performreflection, but with additional mirrors near-normal incidencereflection is possible. The apparatus works by shining broadband light from a variety of sources on a sample. Reflectedor transmitted intensity is measured with a few different de-tectors. A beamsplitter/interferometer apparatus allows theresolution of distinct frequency components. Figure fromhttp://www.pi1.physik.uni-stuttgart.de/research

/Methoden/FTIR e.php. Figure courtesy of N. Drichko.

D. Visible and Ultraviolet

The technique of optical reflection as outlined abovebegins to become more difficult at frequencies on the or-der of the plasma frequencies of metals. To do reflectiv-ity measurements, one always needs a standard sample.Standard metal references cannot be used above theirplasma frequencies as their reflectivity changes quicklywith frequency. Quartz or silicon which have an approx-imately flat (but low) reflectivity can be used as a refer-ence in a grating spectrometer70. However, more com-mon nowadays is to use techniques such as ellipsometryto determine complex optical coefficients in this spectralrange as one can measure complex response functions di-rectly.

Ellipsometry is an optical technique for determiningproperties of surfaces and thin films. Although themethod was originally used as far back as 1887 by PaulDrude to determine the dielectric function of various met-als and dielectrics, it only gained regular use as a char-acterization tool in the 1970s71,72. It is now widely usedin the near infrared (NIR) through ultraviolet (UV) fre-quency ranges in semiconductor processing for dielectricand thickness characterization.

18

When linearly polarized light is reflected from a sur-face at glancing incidence the in- (p) and out-of-plane(s) of incidence light is reflected at different intensitiesas well as suffering a relative phase shift. The reflectedlight becomes elliptically polarized as shown in Fig. 16.The shape and orientation of the ellipse depend on theangle of incidence, the initial polarization direction, andof course the reflection properties of the surface. An el-lipsometer measures the change in the light’s polariza-tion state and characterizes the complex ratio ρ of thein- (rp) and out-of-plane (rs) of incidence reflectivities.The Fresnel equations allow this quantity to be directlyrelated to various intrinsic material parameters, such ascomplex dielectric constant or layer thicknesses.

E

xzysample

plane of incidence

s

p

FIG. 16: (Color) Schematic showing the basic principle ofellipsometry. Glancing incidence light linearly polarized withboth s and p components is reflected with different intensitiesand a relative phase shift. A characterization of the resultingelliptically polarized reflected light’s minor and major axes,as well as tilt angle gives a unique contribution of the complexoptical constants of the material.

The ellipsometer itself is designed to measure thechange in polarization state of the light reflected froma surface at glancing incidence. From a knowledge ofthe orientation and polarization direction of the incidentlight one can calculate the relative phase difference, ∆,and the relative amplitude difference (tanΨ) between thetwo polarization components that are introduced by re-flection from the surface.

Given the complex ratio ρ = rp/rs = ei∆tanΨ of thein- (rp) and out-of-plane reflectivities (rs), various intrin-sic material parameters can be generated via the Fresnelequations. In a homogeneous sample the complex dielec-tric constant ε = ε1 + iε2 is related to ρ as given in Eqs.51, 52 and 53 below. Here as usual ε1 parameterizes thepolarizability of a material, whereas ε2 parameterizes thedissipative properties. εz is the complex dielectric con-stant perpendicular to the reflection surface and εx is thecomplex dielectric constant in the plane of the reflectionsurface, and θ is the angle of incidence as shown in Fig.16.

ρ = rp/rs = tanΨei∆, (51)

rp =

√1− ε−1

z sin2θ −√εxcosθ√1− ε−1

z sin2θ +√εxcosθ

, (52)

rs =cosθ −

√εy − sin2θ

cosθ +√εy − sin2θ

. (53)

In a typical configuration, monochromatic light is in-cident on a glancing trajectory θ close to the Brewsterangle (usually ∼ 65 − 85 for bad metals at high fre-quencies) with a linear polarization state of 45. As thetypical detector measures power and not electric field, itis essential that the light’s elliptical polarization is mea-sured over at least 180 (and more typically 360). Tocompletely characterize the phase, it is clearly not suffi-cient to simply measure the projection of the ellipse alongtwo orthogonal directions, as can be seen by constructionin Fig. 17. In order to get the phase information, the el-lipse’s orientation and major and minor axes must beknown. Generically, the light originally linearly polar-ized at 45 is changed to an elliptical polarization withits major axis displaced from 45. A characterization ofthis ellipse’s major and minor axes, as well as tilt angle isa direct measure of the complex amplitude reflection co-efficients, which can then be related to intrinsic materialquantities.

FIG. 17: (Color) To completely characterize the electric fieldvectors and their phase, it is insufficient to measure the pro-jection of the ellipse along two orthogonal directions. Thecomplete ellipse must be mapped out to get its major andminor axes and tilt angle.

The fact that ellipsometry measures the ratio of twosimultaneously measured values gives it several distinctadvantages as a characterization tool over simple reflec-tivity. It is highly accurate and reproducible even atlow intensities as many systematic errors are dividedout. Moreover, unlike reflectivity measurements, thetechnique is self-normalizing and a reference sample is

19

not necessary. This is important as the choice of a ref-erence can be problematic in the optical and ultravioletspectral range or when the surface is of unknown quality.The method is also particularly insusceptible to sourceintensity fluctuations as they are also divided out. Per-haps most important for the overall utility of the tech-nique is that the method also measures a phase, whichgives additional information regarding materials proper-ties. The phase information can be used to generate thecomplex optical constants (the complex dielectric con-stant for instance, or the complex index of refraction nand k or the complex conductivity σ) which is essentialinformation for a complete characterization of a mate-rial’s optical response. If the optical constants are known,the phase information can be used to sensitively mea-sure films thicknesses. If conventional ellipsometry hasa disadvantage it is that with typical grating monochro-mator data acquisition (i.e. one frequency stepped at atime) it is relatively slow as both the grating and polar-izers must continue to be moved. In such a setup oneloses the rapid acquisition time of the multiple frequencymultiplexing in, for instance, Fourier transform infraredreflectivity (FTIR)9. As mentioned above, the techniqueis well established and widely used in the NIR throughultraviolet (UV) frequency ranges73,74. There have beena number of attempts to extend ellipsometry to lowerinfrared frequencies73–75. Unfortunately the lack of suffi-ciently intense sources (among other reasons) has meantthat such efforts have met with limited success, althoughsynchrotron based efforts have had some success in thisregard76.

The energy scales of the visible and ultraviolet, whichare measured by ellipsometry are not usually explicitlyrelevant for strongly correlated systems. Typically we areinterested in much lower energies that are on the order ofthe temperatures that phenomena are exhibited. How-ever, such energies are relevant for the determination ofimportant band structure parameters on, say, 3D materi-als that photoemission cannot be performed on. Excita-tions in this range also determine the high frequency di-electric constant ε∞, which can have a significant (but in-direct) effect of the low energy physics. Characterizationof optical constants in the visible and UV range is alsoimperative to constrain the phase information at low en-ergies when performing Kramers-Kronig with reflectivitydata. Recently an extremely powerful method for com-bining different data sets using a Kramers-Kronig con-sistent variational fitting procedure was developed by A.Kuzmenko77. It allows a general procedure for combiningsay, DC resistance, cavity microwave measurements at afew distinct frequencies, IR power reflectivity, and ellip-sometry to extract over a broad energy range of all thesignificant frequency and temperature dependent optical

9 In FIR ellipsometers one typically uses the interferometer of anFTIR spectrometer, which retains this multiplexing.

properties.

IV. EXAMPLES

A. Simple Metals

Noble metals like silver, copper, and gold provide agood first step for the understanding the electrodynamicsof solids. Compare the reflection spectra of silver andgold (and aluminum) given in Fig. 18 as a function ofenergy. The reflection spectra of silver shows a sharpplasma edge around 3.8 eV Gold has a much lower plasmafrequency with its plasma edge at a wavelength of around2.5 eV. Also shown is the reflection spectra for aluminumwhich has a much larger plasma frequency than any ofthese at around 15.5 eV. The small dip in Al around 1.65eV is caused by an interband absorption (see below) andis not its plasma frequency.

FIG. 18: (Color) Reflectance curves for aluminium (Al), silver(Ag), and gold (Au) metal at normal incidence as a functionof incident energy. The plasma frequencies of silver and goldare clearly seen. The small dip in aluminum’s curve around1.65 eV is caused by an interband absorption and is not itsplasma frequency. Aluminum’s plasma frequency lies at muchhigher energies at around 15.5 eV. Adapted from Ref. 78.

From these spectra we can immediately see the ori-gin for the visual difference between silver and gold. Asmentioned above, the reflection/transmission propertiesof metals greatly change above their plasma frequencies.If not for the presence of interband transitions, metalswould be transparent in this regime. The lower plasmafrequency of gold is what determines its yellowish color inreflection. It reflects less in the blue/violet portion of thespectra and hence looks yellowish. In contrast the plasmafrequency of silver is in the UV range and so the reflec-tion of silver is almost constant at constant throughout

20

FIG. 19: Real and imaginary dielectric functions for silver(top) and gold (bottom). The thickness of the curve indicatesthe experimental uncertainty, which originates mainly in theKramers-Kroning transform. From Ref. 79.

the infrared to visible range. Aluminum’s plasma fre-quency is higher yet still and so the major differences inits optical properties as compared to silver are invisibleto the human eye.

As discussed above, within the Drude model with only

free carriers the plasma frequency is set by ωp =√

4πNe2

m .

This is frequency of the dip in the reflectivity, the zerocrossing in ε1, and the frequency of free longitudinalcharge oscillations (the plasmon frequency). However,the presence of interband excitations, which contributea high frequency dielectric constant ε∞ changes the sit-uation. In real materials, these features are exhibitedat the so-called screened plasma frequency set by ωp =

√4πNe2

ε∞m . It is interesting to note that the carrier density

and masses of silver and gold are almost identical and sothe difference in their plasma frequencies comes almostentirely from the renormalization effects of different in-terband transitions ε∞.

Data like that shown in Fig. 18 can, with appropriateextrapolations to high and low frequency, be Kramers-Kronig transformed to get the real and imaginary dielec-tric function as shown in Fig. 19 for silver and gold. Asexpected they both show strong almost perfect ω = 0Lorentzian Drude peaks of almost equal weight, but goldshows stronger interband absorptions, gives it a larger ε∞that renormalizes its plasma frequency10. Such spectracan be decomposed into various contributions as shownin Fig. 20 for silver. As expected there is a strong zerofrequency Drude peak contribution as well as interbandcontributions. Although the interband piece shows someresemblance to the Drude-Lorentz plots in Fig. 7, theshape is a more rounded cusp, which is characteristic oftransitions within the manifold of d-electron states.

10 Recall from section V B that a low frequency oscillator (absorp-tion) can give a large ε∞. All other things being equal, the lowerthe frequency of the oscillator the higher the ε∞.

21

FIG. 20: Decomposition of the dielectric function of silverinto Drude and interband contributions. The threshold ab-sorption for interband excitations is indicated by ωi. ε

(1)1 is

the Drude contribution. δε1 is the interband piece. From Ref.81.

In Fig. 21 a similar plot of the complex response func-tions for aluminum is shown. In addition to the metallicω = 0 Lorentzian Drude peak, the prominent absorptionthat gives the dip in the reflectivity curve around 1.65 eVcan be clearly seen. That this dip in the reflectivity doesnot reflect the plasma frequency can be see in the plotof the loss function Im (− 1

ε ), which shows a strong peakonly at the ‘screened’ plasma frequency 15.5 eV (i.e. thezero cross of ε1.

B. Semiconductors and Band Insulators

In semiconductors and band insulators, the real partof the conductivity is dominated by interband transitions(red in Fig. 2). At low temperatures in a clean undopedinsulator, naively one expects to see a gap with no con-ductivity and then a sudden onset at the gap edge. Infact, various other contributions are also possible insidethe gap. Electron-hole pair bound states can form as ex-citons and give dissipative response and be seen as sharppeaks below the gap. This is found for instance in Fig.22 for the ionic insulator KCl, where in addition to the

FIG. 21: (top) Real and imaginary dielectric functions foraluminum. (bottom) Real part of the conductivity and theloss function. In addition to the Drude peak, a prominentabsorption is found around 1.65. This feature gives the dip inthe reflectivity curve around the same energy. The loss func-tion shows a strong peak at the screened plasma frequency15.5 eV. From Ref. 80.

band edge excitation there are a series of below gap ex-citon absorptions. Even lower measurement frequenciesthan those shown would reveal a series of sharp peaksfrom optical phonons down in the 50 meV range. Suchbehavior can be seen in Fig. 23 for MgO and FeSi.

If one starts to dope into a typical semiconductor likephosphorus into silicon, massive changes in the opticalspectra occur. At very low doping levels one can discernthe sharp absorptions of individual donors as seen in thebottom of Fig. 24. As the dopant density is increasedinter-pair and inter-cluster quantum tunneling broadensthe spectra until a prominent a prominent impurity bandforms in the gap as shown in Fig. 25. Charge transitionscan occur in this manifold of impurity states. Below acertain critical doping such a doped state retains its in-sulating character and is termed an electronic glass. Idiscuss this state of matter below.

C. Electron glasses

Many crystalline and amorphous semiconductors showa metal-insulator transition (MIT) as a function of

22

FIG. 22: Imaginary dielectric function of KCl. The absorp-tion around 9 eV is the bad gap. The series of sharp featuresat energies below the band gap are due to excitons. FromRef. 82

dopant concentration or stoichiometry. On the insulat-ing side of the transition, the materials are insulators notprincipally for the reason of completely filled bands (asin the case of a band insulator) or interactions (as in thecase of a Mott insulator), but because electronic statesat the chemical potential are localized due to disorder.Such materials which have a random spatial distributionof localized charges have been called - in analogy withstructural glasses - electron glasses and are more gener-ally termed Anderson insulators.

At finite temperatures these materials conduct underDC bias via thermally activated transport of chargeshopping between localized states in an impurity band.Hence, the DC conductivity tends to zero as the tem-perature tends to zero. However, because their insulat-ing nature derives from localization of the electronic or-bitals and not necessarily a vanishing density of statesat EF , such materials can have appreciable conductivityat low frequency deriving from transitions between local-ized states inside the impurity band. Although electrontransport in electronic glasses is still not completely un-derstood, different regimes for AC transport can be dis-tinguished depending on the strength of electron-electroninteractions 11, the energy scale being probed and the

11 There has been a lot of recent interest in the subject ofmany-body localization. The proposal of Basko, Aleiner, andAltshuler87 is that a disordered interacting localized system can-not function as its own heat bath. In other words, if the roleof delocalized states like phonons could be dispensed with thensuch a system would be completely insulating even at finite tem-

FIG. 23: Optical conductivityof MgO (top panel) and FeSi atT=4K (bottom panel) showing the strong phonon absorptionfound in such insulating compounds. In the insets are theintegrated spectral expressed in units of effective charge (SeeSec. V A), with the electronic contribution removed. ForMgO the effective measured transverse charge is 1.99, whichis in good agreement with the formal valences of the Mg+2 andO−2 ions. From Ref. 83. For further discussion on effectivecharge see Ref. 18.

amount of disorder. See Helgren et al.86 for a completediscussion on this subject.

Far away from the MIT and in regimes when electron-electron interactions are not significant, one considersa so-called Fermi glass. This is an ensemble of local-ized charges whose properties are primarily determinedby Fermi statistics. When the long-range Coulomb in-teraction is of the same order as the disorder poten-tials, a so-called Coulomb glass emerges, and finally,near the transition to the metallic state, fluctuationslead to a quantum critical regime. For these variousregimes, certain characteristic power law and exponen-tial functional forms for the AC and DC transport areexpected. For instance, in a Fermi glass the AC con-

perature. Signatures of these effects have been inferred in coldgases88 and it has been proposed that if coupling to phonons isweak then there still may be some signatures of the finite temper-ature insulating phase89. It has been proposed the AC conduc-tivity may hold important signatures of these effects90. There isessentially no experimental work here at all looking into this inthe solid-state, but the subject is tremendously interesting. Seerecent reviews for more details91.

23

FIG. 24: Absorption coefficient in P doped Si normalizedto the number of donors nD for the three different labeleddoping levels at 2 K. At the lowest doping, absorptions dueto individual donor atoms can be seen. These absorptionbroaden into a impurity band at higher dopings. The directgap is at much higher energies as shown in Fig. 25. FromRef. 84

FIG. 25: Optical conductivity of silicon doped close to the3D metal-insulator transition. The optical band gap is visiblevia the high energy onset at 1.12eV. The intermediate energyconductivity is associated with excitations from the dopantband to the conduction band. The conductivity at the lowestenergies, i.e. photon assisted hopping conductivity, is due tointra-dopant band excitations. From Ref. 86.

ductivity is expected to be a power law σ ∝ ω292. In aCoulomb glass the conductivity is expected to take theform σ ∝ U(r) · ω + ω293,94. Here, the non-interactingFermi glass functional form is returned at high frequen-cies, but a linear dependence is found at low frequencies.The crossover between these regimes is smooth and set by

U(r) which is the typical long range interaction strengthbetween the charges forming the absorbing resonant pair.There are expected to be logarithmic corrections to thesepower laws. As the energy scale U(r) characterizes thelong range Coulomb interaction, it is expected to go tozero at the MIT. As one approaches the quantum criticalpoint, the behavior may also be interpreted using scal-ing laws (discussed below) that connect the temperature,frequency and concentration dependence of the response.It is expected that there is a correspondence between thefrequency and temperature dependent conductivity onboth sides of the critical concentration. Such an analysisof the conductivity leads to a universal scaling functionand defines critical exponents as discussed below.

In Fig. 26, the pioneering data of Stutzmann and Lee51

on Si:B demonstrates the real part of the conductivitymeasured with a broad band microwave Corbino spec-trometer. The data shows a remarkable observation ofconductivity linear in frequency at low frequencies and asharp crossover at U(r) to ω2 at high frequencies. Workon Si:P on samples farther from the metal-insulator tran-sition shows the same behavior85,86, albeit the energyscale of the crossover is found at higher energies as itis set by the long range Coulomb interactions (whichincreases away from the MIT). Although the proposedfunctional form of Efros and Shklovskii approximatelyfits the data of Stutzmann and Lee51 (and of Helgrenet al.85), it fails to account for the very sharp crossoverbetween power laws. Stutzmann and Lee51 interpretedthe sharp crossover as deriving from a sharp feature inthe density of states - the Coulomb gap (a depression atthe Fermi energy in the density of states that is a conse-quence of the long range Coulomb interaction). However,such an interpretation is at odds with theory and more-over, was not consistent with the work of Helgren et al.85,where it was shown that the crossover was set by the typ-ical interaction strength. The sharp transition betweenpower laws remains unexplained; although one may con-sider that this sharp crossover may arise from collectiveeffects in electron motion96.

The theoretical functional form and experimental ob-servation is remarkable, because the ordering of regimesis at odds with the typical situation in solid state physics.Typically it is that non-interacting functional forms areexpected at low frequency and temperature, while theexplicit effects of interactions are seen at higher fre-quencies. The canonical example of this ordering is theFermi liquid, where it is the low frequency spectra whichcan be typically modeled in terms of quasi-free non-interacting particles (perhaps with renormalized masses).The higher energy spectra are typically more complicatedand show the non-trivial effects of electron-electron andelectron-boson interactions. Similar physics is exhibitedin the heavy-fermion compounds in that it is the low fre-quency regime that can be understood in terms of con-ventional Boltzmann transport of heavy electrons. Incontrast, in electron glasses, it is the low frequency spec-tra which shows the effect of interactions while the high

24

FIG. 26: The real part of the complex frequency dependentconductivity for two samples of Si:B at 85% and 88% of theway to metal-insulator transition51. Data has been measuredup to 20 GHz with a Corbino geometry microwave probe. Thecrossover in the frequency dependent conductivity from linearto quadratic is qualitatively consistent with a crossover fromCoulomb glass to Fermi glass behavior.

frequency spectra returns the non-interacting functionalform.

This was not the original expectation. Anderson orig-inally coined the term “Fermi glass” in analogy withthe Fermi liquid, to describe a ensemble of localizedcharges whose properties where largely determined byFermi statistics alone97.

"(The) Fermi liquid theorem is a rigorousconsequence of the exclusion principle, ithappens because the phase space availablefor real interactions decreases so rapidly(as E2 or T2). The theorem is equally truefor the localized case: at sufficientlylow temperatures or frequencies thenon-interacting theory must be correct,even though the interactions are notparticularly small or short range: thus thenon-interacting theory is physically correct:the electrons can form a Fermi glass."

- P.W. Anderson, 1970

It is surprising that even in a material as thoroughlystudied as doped bulk silicon that there exists no clearconsensus as to the ground state and the nature of thelow energy excitations at low dopings. It appears how-ever, that Anderson’s speculation is not borne out by theexperimental situation. The low lying excitations appearfundamentally changed by interactions. Of course, ex-isting experiments do not preclude that at even lowerenergies the Coulomb glass-like behavior will not breakdown or give way to a different response like a Fermiglass.

D. Mott insulators

Mott insulators differ from band insulators and Ander-son insulators in that their insulating effects derive fromcorrelations and not from filled bands or disorder-drivenlocalization. One expects to see very interesting aspectsof spectral weight transfer from high energies to low ascharges are doped into them and they eventually becomemetals.

At half-filling it is believed that such Mott insulatorsare characterized by upper and lower Hubbard bands,which are split by an energy U that is the energy to dou-bly occupy a single site. If one wants to occupy a sitedoubly, one must pay this energy cost U . As will be dis-cussed below the spectral weight

∫σ(ω)dω is a conserved

quantity which depends on the total amount of charge ina system. It is interesting to account for how spectralweight is transfered from high to low energies by dopingelectrons into such a system98.

At half-filling, the lower Hubbard band is filled (oc-cupied) and the upper Hubbard band is unfilled. Sinceevery site in the system has one electron, the spectralweight of upper and lower Hubbard bands are equal. Nowconsider the situation after one dopes a single electronto the upper Hubbard band. The unoccupied spectralweight of the upper Hubbard band reflects the total num-ber of sites which can have an electron added, the lowerHubbard band reflects the total number of sites occupiedby a single electron, and the occupied weight of the upperHubbard band reflects the total number of sites that aredoubly occupied. These considerations then mean thatdoping a single electron transfers one state each fromboth upper and lower Hubbard bands as shown in Fig. 27because doping changes the number of sites which are oc-cupied/unoccupied. It is natural to expect that a state isremoved from the unoccupied part of the upper Hubbardband as now a state is occupied. However a state is alsoremoved from the occupied part of the lower Hubbardband as its spectral weight quantifies how many elec-trons sit on singly occupied sites (and hence how manycan be removed from singly occupied sites). The occupiedweight of the upper Hubbard band is then ‘2’, and thespectral weights of occupied lower Hubbard band andunoccupied upper Hubbard band are both N-1. The sit-uation differs from doping a band semiconductor becausethen doping a single electron into the upper band, onlytakes that single state from the upper band. There is notransfer of spectral weight across the gap. An interme-diate case is expected for the charge-transfer insulators,which the parent compounds of the high Tc supercon-ductors are believed to be. In these compounds a chargetransfer band (consisting primarily of O 2p states) sitsin the gap between upper and lower Hubbard bands andplays the role of an effective lower Hubbard band. Theenergy to transfer charge from from an oxygen to cop-per ∆ becomes the effective onsite Hubbard U . Issues ofspectral weight transfer in Hubbard and charge transferinsulators are discussed more completely in Ref. 98.

25

FIG. 27: A schematic of the electron-removal and additionspectra for a simple semiconductor (left), a Mott-Hubbardsystem in the localized limit (middle) and a charge transfersystem in the localized limit. (a) Undoped (half-filled) (b)one-electron doped, and (c) one-hole doped. The onsite re-pulsion U and the charge-transfer energy ∆ are indicated.From Ref. 98.

In Fig. 28 we show the doping dependence of the roomtemperature optical conductivity of La2−xSrxCuO4. Atx = 0 the spectra show a clear charge transfer gap ofabout 1.8 eV. This largely reflects excitations from O2p6 states to Cu 3d10. Upon doping with holes, spectralweight is observed to move from the high energies to lowin a manner consistent with the picture for the chargetransfer/Mott Hubbard insulators in Fig. 27. It is in-teresting to note that a remnant of the charge transferband remains even for samples that have become super-conductors and are known to have a large Fermi surface.

E. Superconductors and other BCS-like states

One can get a rough intuition of the electrodynamicresponse of superconductors for ~ω 2∆ and TTcfrom the relations in Eq. 25, which give the dissipation-less limit for the Drude model. One expects a δ functionpeak at zero frequency in the real conductivity and a1/ω dependence in the imaginary part, whose coefficientis set by the strength of the δ function. At frequencies onthe order of the superconducting gap 2∆, Cooper pairscan be broken and enhanced absorption should be found.At higher temperatures, thermally excited quasiparticlesare created and one expects that as they are subject toessentially normal state dissipative processes, they willgive a contribution to σ1 at finite ω.

Very roughly one expects a two fluid scenario wherethe conductivity is approximately given by σ(ω) =π2Ne2

m δ(ω = 0)+iN(T )e2

mω +σ1n(ω, T ) where σ1n is a normalfluid component. At low temperatures, one expects thatthe superfluid density N(T ) is degraded as 1− e−∆/kBT ,as Cooper pairs thermally disassociate. As the superfluiddensity decreases the normal fluid component increasesand one may in some circumstances expect an increasein σ1n to go as e−∆/kBT .

FIG. 28: Doping dependence of the room temperature opticalconductivity of La2−xSrxCuO4. At x = 0 the spectra showa clear charge transfer gap of about 1.8 eV. Upon dopingwith holes, spectral weight is observed to move from the highenergies to low in a manner consistent with Fig. 27. Figureadapted from99.

.

These above considerations are only approximate how-ever. The explicit temperature dependence of the gapand the mutual screening effects of the superfluid andnormal fluid must be taken into account. Among otherthings, this means that the functional form of the contri-bution from ‘normal’ electrons σn, although still peakedat ω = 0, will be decidedly non-Lorentzian. Within thecontext of the BCS theory the response of a superconduc-tor can be calculated from the Mattis-Bardeen formalism.I will not go into the details of the calculation here. In-terested readers should consult Tinkham16. One of theimportant results from this formalism however is the ef-fect of the superconducting coherence factors, which de-pending on their sign can lead to enhanced or suppressedabsorption over the normal state above and/or below thegap edge. This means that different symmetries of thesuperconducting order parameter (or order parametersof other BCS-like condensates like spin- or charge-densitywaves), leave their signatures on the dissipative response.

Shown in Fig. 29 is the temperature and frequencydependence of the real part of the conductivity σ1 forω > 0 as evaluated from the Mattis-Bardeen expressionfor ‘type II’ coherence factors, which are appropriate fors-wave superconductivity. One can see that at T=0, onehas no below gap absorption, and then a gentle rise be-gins starting at the gap edge 2∆. Despite the singularityin the density of states, the conductivity is suppressed at

26

FIG. 29: Temperature and frequency dependence of theconductivity σ1 as evaluated from the Mattis-Bardeen ex-pression. The coherence peak exists only at low frequencies~ω/2∆ < 0.1. The cusp in the surface corresponds to theenergy gap. From Ref. 102.

the gap edge, as the type II coherence factors effectivelycancel the enhanced density of states. As one warmsthe sample, below gap excitations are possible from ther-mally excited quasiparticles. Just as above the gap edgethe absorption is suppressed, the absorption is enhancedover the normal state at energies below the gap. This isagain the consequence of type II coherence factors.

This anomalous behavior is more clear along the tem-perature axis. If one follows the conductivity at a par-ticular frequency as a function of temperature, one willsee an enhanced absorbence at temperatures below Tc.This result although surprising is direct evidence for co-herence effects in the superconductor. An analogous en-hancement found in the nuclear relaxation rate via NMR- the so-called Hebel-Schlichter peak - was early evidencefor the BCS theory100.

These features contrast with the situation in Type Icoherence factors as shown in Fig. 30. In a Type I co-herence factor SDW material one expects a very differentresponse. There the coherence factors don’t cancel thesingularity in the density of states. Likewise, there is noHebel-Slichter-like peak in the temperature dependence.

Qualitatively the behavior for Type II coherence fac-tors is realized in conventional s-wave superconductors.In Fig. 31, the normal state normalized conductivityfrom some of the original measurements of Palmer andTinkham101 is shown, which is in good agreement withthe Mattis-Bardeen prediction. The suppression at thegap edge is consistent with type II coherence factors andhence a s-wave superconducting state. Mattis-Bardeenpredicts that subgap absorptions rise with increasingtemperature and become enhanced over the normal stateconductivity.

In Fig. 32 the conductivity of Nb at 60 GHz as mea-sured in a microwave resonance cavity is shown as a func-

FIG. 30: (Color) (left) Approximate frequency dependenceof the dissipation rate from the Mattis-Bardeen theory forType I and Type II coherence factors. (right) Approximatetemperature dependence of the dissipation rate for ω = 0.1∆for both coherence factors.

FIG. 31: Frequency dependence of the normalized conductiv-ity through lead films at three different temperatures. Theresults are obtained through a combination of reflection andtransmission measurements. The solid line is a calculationusing the Mattis-Bardeen theory with a 2∆/~ = 22.5 cm−1.Tc for these lead films is 7.2 ± 0.2 K. From Ref. 101.

tion of temperature102. The peak below Tc is the electro-magnetic equivalent of the Hebel-Slichter peak in NMR.This is again consistent with s-wave superconductivity.

As noted, different behavior is expected for type I co-herence factors. There one expects an absorption en-hancement at the gap edge and no ‘Hebel-Slichter’ peak.

27

FIG. 32: Temperature dependence of the complex conduc-tivity σ(ω) of Nb as evaluated from surface impedance mea-surements at 60 GHz in cavities. The solid curve is the weakcoupling Mattis-Bardeen prediction. The dashed curve is astrong coupling Eliashberg prediction. From Ref. 102.

The lack of such a peak in electromagnetic absorption ofcuprate superconductors is at least partial evidence ford-wave superconductivity in those compounds. It is duein part to the lack of a strong singularity in the d-wavedensity of states, but also because the coherence factorvanishes for q = π, π since ∆k ≈ −∆k+π,π for k nearπ, 0. Such data is shown in Fig. 33 where the conduc-tivity measured at a number of different frequencies inthe GHz range on YBCO is displayed as a function oftemperature103. The data shows a broad peak in σ1 ataround approximately 35 K, but a comparison with theMattis-Bardeen prediction shows that it is incapable ofdescribing it as the rise in σ1 is much more gradual thanpredicted. An explicit comparison is shown in Fig. 34.This peak has been quite reasonably described in termsof a collapsing scattering rate below Tc of quasiparticleswhose number remains appreciable until low tempera-tures due to the d-wave nature of this compound. Anexamination of the Drude model shows that in generalone expects a peak in σ1 when the probe frequency is ap-proximately equal to 1/τ the scattering rate. The exhib-ited peak is therefore not a Hebel-Slichter peak, and thedata is consistent with d-wave superconductivity. Theseexperiments give evidence for d-wave superconductivitytwice: the lack of a Hebel-Slichter peak and the large σ1

due to quasiparticle effects. Note that the other sharppeak right at Tc is believed to be due to fluctuations ofsuperconductivity. Its functional form is also inconsistentwith it being a Hebel-Slichter peak.

For completeness, I should mention that spin densitywave compounds which can be treated within the BCSformalism are expected to have a type I order parameterand a gap edge absorption enhancement. In many casesthey can also be treated within a BCS formalism. SeeRef. 105 for explicit details.

FIG. 33: The real part of the conductivity extracted fromthe microwave surface resistance of YBCO at a number ofdifferent GHz frequencies. From103.

FIG. 34: The real part of the conductivity extracted from themicrowave surface resistance of YBCO. The BCS conductivity(solid) is calculated using a Tc of 91.8 K, a gap ratio of 3.52,and various other physical parameters such as penetrationdepth, coherence length, and mean free path. It is completelyincapable of describing the EM response, which shows the d-wave nature of these materials. From Ref. 104.

A vast literature exists on the electrodynamics of su-perconductors. I have given only the most superfi-cial treatment here. Many considerations go into howsuperconductivity is exhibited in optical spectra. SeeTinkham16, for instance, for in-depth discussions on theclean and dirty limits of superconductivity and Basov andTimusk20 for the state-of-the-art on high-temperaturecuprate superconductivity.

28

V. ADVANCED ANALYSIS AND TECHNIQUES

A. Sum Rules

The optical constants satisfy a number of different sumrules14. Although a number of different ones can be de-fined (see below), the most frequently used concerns thereal part of the conductivity. As alluded to above, thereis a relation for the integral over all frequencies of the realpart of the conductivity to the total number of chargesn and their bare masses me.

8

∫ ∞0

σ1(ω)dω =4πne2

me. (54)

Analysis of data in terms of sum rules provides a pow-erful tool that can be used to study spectral weight dis-tributions in a relatively model-free fashion. The aboveintegral which extends from zero to infinity is the globaloscillator strength sum rule, which relates the integral ofthe σ1 to the density of particles and their bare mass. Itis related to the oscillator strength fs,s′ discussed abovein the context of the Kubo formula. Both are manifesta-tions of the f -sum rule of elementary quantum mechan-ics. Sum rules follow as consequence analyticity of theresponse functions and causality. See Refs.12,14 for a fewderivations.

FIG. 35: The effective number of carriers neff (Ωc) as a func-tion of cutoff frequency Ωc for aluminum. A number of differ-ent sum rules are investigated and displayed. The sum rule forε2 is equivalent to the sum rule expressed in Eq. 54. Figureadapted from Ref. 106.

In real systems the integral to infinite frequencies israrely utilized in practice and we are usually more con-cerned with partial sum rules such as

8

∫ W

0

σ1(ω)dω =4πne2

mb. (55)

where W is the unrenormalized electronic bandwidth andmb is the band mass. Sum rules of this fashion can be

used to get information about occupation and correlationin a subset of bands. Of course if the integral cut-off Wis extended to high enough energies the full sum rule Eq.54 must be fufilled This puts a fundamental constrainton the various values that the individual sub-band bandmasses can assume. A partial sum rule analysis of theequivalent quantity ε2 applied to the simple metal caseof aluminum is shown in Fig. 35. A number of differentsum rules are also displayed. Aluminum’s nominal elec-tronic configuration is 1s22s22p63s23p1. Depending onthe cutoff of the integral, the partial sum rule for differ-ent orbitals is satisfied. Integrating up to approximatelythe plasma frequency (15.5 eV) returns the number ofelectrons in the n = 3 valence band. At the x-ray L edgeat around 100 eV, one begins to reveal spectral weight ofthe n = 2 formed bands. One can see that if the integralis performed to high enough energies (well above the Kedge of aluminum) one recovers the aluminum’s atomicnumber ‘13’ if one assumes the free-electron mass in Eq.55. It is also interesting to note that the rate of con-vergence is different for the different sum rules plotted.As will be discussed briefly below, this is related to howspectral weight is distributed in the spectra and can berelated to quantities like the strength of interactions.

In correlated systems one frequently makes use of aneven more limited sum rule and only performs the inte-gral over an energy only several times the Drude widthto determine the mass renormalized due to interactions.For instance the small spectral weight in the very nar-row Drude peak as shown in Fig. 8 gives a very largemass (≈ 102me). An integral as such over a small renor-malized Drude peak with small spectral weight is howthe very large masses in Fig. 5 were generated. Onecan view this small mass as coming from a very narrowinteraction derived band where the integral only has tobe performed over a small fraction of W to capture itsspectral weight. However, the sum rules of Eqs. 54 and55 must ultimately be satisfied at high enough energies.This means that there must be be some higher energysatellite in σ1 that contains enough spectral weight toreturn the band mass mb when the integral in Eq. 55is performed out to the unrenormalized band width W .Note that similar information can be gained from apply-ing the extended Drude model analysis, which is outlinedin the next section (Sec. V B).

Sum rules have been used extensively in the analysis ofthe data for correlated electron systems because, in somecircumstances, they allow a relatively model independentway to analyze the data. For instance, one can show thatfor a tight binding model with nearest neighbor hoppingonly that

8

∫ W

0

σ1(ω)dω = −πe2ar

2~2Kr. (56)

where ar is the lattice constance in the incident E field’spolarization direction and Kr is the effective kinetic en-ergy. This relation has been used extensively in the

29

cuprate superconductors to attempt to show evidence foror against a novel lowering of the kinetic energy-drivenmechanism for superconductivity. This is in contrast tothe usual mechanism in the BCS theory where it is thepotential energy that drops when falling into the super-conducting state. Such analysis is not trivial in thesematerials, as there is evidence that the spectral weightthat goes into the condensate is derived not just from thenear EF electronic band over a width much smaller thanW , but over an energy range many times the band widthW 12.

FIG. 36: Temperature dependence of the low-frequency spec-tral weight A1+D(T ) and the high-frequency spectral weightAh(T ), for optimally doped and underdoped Bi2212. Insets:derivatives −T−1dA1+D/dT and T−1dA1h/dT . From Ref.107.

It is believed that the usual sum rule for supercon-ductors (Ferrell-Glover-Tinkham)16 is satisfied for the abplane conductivity of the cuprate superconductors up toabout 10% accuracy20. Molegraaf et al.107 used a com-bination of reflectivity and spectroscopic ellipsometry inBi2212 (Fig. 36) to conclude that about 0.2 - 0.3% ofthe total strength of the superconducting δ function iscollected from an energy range beyond 10,000 cm−1 to20,000 cm−1 13 and that the kinetic energy decreases in

12 It may be that this extreme mixing of low and high energy scalesbetween, which prohibits a true low energy description is an al-ternative definition of a correlated material.

13 10,000 cm−1 is approximately equivalent to 1.24 eV and 14,390K! It is remarkable that such energy scales are explicitly relevantfor a 100 K phenomenon. In contrast conventional superconduc-tors satisfy the ‘Ferrell-Glover-Tinkham’ sum rule at energiesapproximately 10 - 20 ∆16.

the superconducting state. This is a very small effect,but still large enough to account for the condensation en-ergy. Such analysis of total spectral weight transfer canbe tricky as it requires the analysis of spectral weight be-yond the measurement regime, which is only constrainedthrough the Kramers-Kronning relations. These resultsor rather the analysis of the data has been disputed byBoris et al.,108 who form a nearly opposite conclusion re-garding spectral weight transfer based on nearly identicalexperimental data.

The c-axis response has been investigated in a simi-lar manner and a similar transfer of spectral weight fromvery high energies to the δ function has been observed,although in this case the magnitude of the effect is defini-tively insufficient to account for Tc

20.As mentioned above, there are a number of sum rules

in addition to the most commonly applied one for σ1 (orequivalently the first moment sum of ε2). Probably nextmost commonly applied is the sum rule for the first mo-ment of the loss function, whose integral is equal to thatfor the first moment sum of ε2

14. This has been called thelongitudinal f -sum rule. A similar sum rule concerns thefirst moment sum of the absorption coefficient κ(ω)106.There is also a sum rule on the loss function itself, thatcan be related to the Coulomb correlation energy of theground state wavefunction109,110. Similarly, there is sec-ond moment conductivity sum rule and a third momentloss function sum rule that in certain circumstances al-lows one to evaluate the strength of umklapp scatteringand the static electron-lattice energy111,112. There arealso first inverse moment sum rules for the conductivityand loss function, which are equal to various combina-tions of pure numbers (For instance

∫∞−∞−Im 1

ωε(ω)dω =

π). These allow the calibration of, for example, the ab-solute intensity of an electron energy loss measurementsince these sums have no dependence on any materialparameters. Other related sum rules include the inertialsum rule on the index of refraction (

∫∞0

[n(ω) − 1]dω =

0)113–115 and the “DC sum rule” on the dielectric func-tion (

∫∞0

[ε1(ω) − 1]dω = −2π2σDC)113,114. There isalso a sum rule on the frequency dependent scatteringrate136, which is discussed below. For further detailson sum rules, see the very extensive literature on thissubject14,18,106,109–115,136.

B. Extended Drude Model

Within the Drude-Lorentz (Sec. II B ) model we con-sider that conduction electrons are quasi-free. The scat-tering rate of the Drude intraband contribution is con-sidered to be characterized by a single frequency inde-pendent relaxation time τ . It is well known however thatvarious inelastic channels can contribute to relaxation insolids, each with a characteristic frequency dependence.For instance, in 3D the fermion-fermion scattering rate isexpected to exhibit an ω2 dependence at low frequency.An electron scattering off an Einstein boson (a single

30

spectral mode with a well-defined frequency ω0) is ex-pected to show an onset in increased scattering at theboson frequency ω0 which is the threshold to make a real(as opposed to virtual) excitation.

A number of authors have proposed various treatmentsthat can capture these frequency dependences in opticalspectra. Gotze and Wolfle introduced a memory functionM(ω) approach to capture correlations effects where theoptical conductivity could be written in a modified Drudeform as

σ(ω) =ω2p

4π(M(ω)− iω). (57)

Similar ideas around the same time for a frequency de-pendent scattering rate were introduced by P.B. Allenin the context of the electron-phonon problem117. J.W. Allen and J. C. Mikkelsen118 proposed that onecould capture frequency dependent effects experimentallythrough an extended Drude model (EDM) where themass and scattering rate are explicitly frequency depen-dent. Inverting the complex conductivity one gets

m∗(ω)

mb= −

ω2p

4πωIm

[1

σ(ω)

], (58)

1

τ(ω)=ω2p

4πRe

[1

σ(ω)

], (59)

where mb is the band mass.Both in spirit and formalism this is reminiscent of

standard treatments in many-body physics where oneposits that the effects of interactions can be capturedby shifting the energy of an added electron from thebare non-interacting energy ε(k) by a complex self -energy Σ(ω,k) = Σ1(ω,k) + iΣ2(ω,k). One can expressthe optical response in terms of the complex self-energyΣop(ω) = Σop1 (ω) + iΣop2 (ω) as

σ(ω) = −iω2p

4π(2Σop(ω)− ω). (60)

and in terms of previously defined quantities

Σop1 =ω

2(1−m∗/mb),

Σop2 = − 1

2τ(ω). (61)

Note that as 1τ(ω) and m∗(ω) are a particular parame-

terization of the real and imaginary parts of a complex re-sponse function, they are Kramers-Kronig related. Anal-ogous to the case of σ(ω) detailed above, if one knows

1τ(ω) for all ω, then m∗(ω) can be calculated and vice

versa. In a related fashion, if there are frequency depen-dent structures in 1

τ(ω) then the effective mass must be

enhanced in some frequency regions.

A caveat should be given about the interpretation ofoptical self-energies. Although an analogy can be madeto a momentum averaged version of the quasi-particleself-energies that appear in one-particle spectral func-tions, the two self-energies are not exactly the same.While quasi-particle scattering depends only on the to-tal charge scattering rate to all final states, transportand optics experiments have a much larger contributionfrom backward scattering than forward scattering as itdegrades the momentum much more efficiently. In gen-eral this means that the quantities derived via optics em-phasize backward scattering over forward scattering andso the optically derived quantities can contain a vertexcorrection not included in the full quasi-particle inter-action. In this sense the two self-energies are formallydifferent although they can contain much of the sameinformation. The connection in so far as it is under-stood explicitly is discussed in Ref. 119. A number ofother different methods exist to calculate the finite fre-quency scattering rate116,120–122 and a nice comparativetreatment of a number of different approaches has beenrecently given123.

As mentioned above, frequently the extended Drudemodel can be introduced in terms of a complex frequencydependent Memory function, M(ω)

σ(ω) =ω2p

4π(M(ω)− iω)(62)

The memory function M(ω) can be seen defined interms of an effective scattering rate and effective mass1/τ(ω) − iωλ(ω), where τ is the lifetime and 1 + λ =m∗/mb and hence

σ(ω) =ω2p

4π(1/τ(ω)− iω[1 + λ(ω)]). (63)

One can put Eq. 63 in the form of the standard Drudemodel with the substitutions τ∗(ω) = [1 +λ(ω)]τ(ω) andω∗2p = ω2

p/[1 + λ(ω)] to get

σ(ω) =ω∗2p (ω)

4π

τ∗(ω)

1− iωτ∗(ω). (64)

This equation describes the frequency dependence of aparticle with renormalized plasma frequency (and hencerenormalized mass) and renormalized scattering rate.Compare Eq. 64 with Eq. 26. Note that the quantity1/τ∗ is not equivalent to the quantity defined in Eq. 59above, as 1/τ∗ includes the renormalization effects of thelifetime as well as the mass, whereas 1/τ includes onlylifetime effects. The two quantities differ by a factor of1 + λ(ω). In this sense one may argue that 1/τ(ω) is themore intrinsic quantity, as also evidence by its direct pro-portionality to the imaginary part of the self-energy, butthis point is essentially a semantic one. 1/τ∗ is the effec-tive scattering rate (or dressed scattering rate) however,

31

FIG. 37: Scattering rate vs. ω2 of (a) Cu, (b) Ag and (c) Au. Solid and dotted lines are the bare scattering rate 1/τ and thedressed scattering rate 1/τ∗ respectively as detailed in the text. From124

FIG. 38: Mass enhancement factor 1 + λ of noble metals.From124

because for weak frequency dependence it is the actualhalf width of the Drude peak in the optical conductivity.

Various contributions to the total scattering rate suchas electron-electron (ee), electron-phonon (ep), electronmagnon (em), etc. add within the prescription set byMatthiesen’s rule as 1

τ(ω) = 1τ0

+ 1τee

+ 1τep

+ 1τem

. Note

that there is typically also a contribution 1τ0

to the elas-tic scattering which is frequency independent that comesfrom static disorder. As mentioned above this can beseen as a measure of the degree to which the translationalsymmetry is broken by disorder and hence the degree towhich strict momentum conservation in the optical selec-tion rule q = 0 can be violated.

The near-EF electrons of the noble metals, like Cu,Ag, and Au have free-electron-like s-states, which forma Fermi surface that deviates from almost spherical sym-metry only near the Brillouin zone necks in the < 111 >directions. The above analysis applied to such metals isshown in Fig. 37 and Fig. 38. For Cu and Ag the scat-tering rate shows an approximately ω2 dependence of thescattering rate, which is in accord with the expectation

FIG. 39: The quantity 1/τ∗(ω) from Eq. 63 for NiAs andNiSb118. One can see that below the energy scale of the in-terband transitions (0.4 eV and 0.475 eV) an approximatelyω2 dependence is found.

for electron-electron scattering 14. It is also interestingthat electron-electron scattering is observed up to suchhigh energies as we may have expected other contribu-tions such as electron-phonon scattering to give an ob-servable contribution. We should reiterate that a purelytranslational invariant electron gas, cannot dissipate mo-mentum by electron-electron collisions alone. In this re-

14 One should interpret the reporteded ω2 dependence here for sim-ple metals carefully. Indeed one may expect that electron-phononscattering would be more dominant in simple metals.

32

gard effects like umklapp or interband (Baber) scatteringare essential to see the effects of such collisions in the op-tical response. Also note here the difference between thebare scattering rate 1/τ and the dressed scattering rate1/τ∗. The difference is only significant when there is alarge renormalization in the effective mass.

In Fig. 39 are shown extended Drude model datafor the quantity 1/τ∗(ω) from Eq. 63 for NiSb andNiAs118. These antimonides are more strongly interact-ing 3d systems. The room-temperature optical reflectiv-ity of metallic NiSb and NiAs were measured between0.05 and 5.0 eV, and the optical constants have been de-duced by a Kramers-Kronig analysis. Unlike the goodmetals discussed immediately above, the color of thesesystems is more dull with NiAs appearing yellow, andNiSb appearing light pink due roughly to the same mech-anism as gives copper and gold its colors. One can seein Fig. 39 that below the energy scale of the interbandtransitions (0.4 eV and 0.475 eV) an approximately ω2

dependence is found. This is consistent with the Fermiliquid expectation. However notice the vastly larger scaleof the coefficient of the ω2 term in as compared to thesimple metals.

Although the dual quadratic dependencies of the ω2

and T 2 forms of the electron-electron scattering rates ina Fermi liquid follows in a relatively model independentfashion, more consideration needs to go into their relativemagnitudes. Indeed, there has been recent interest in therelative prefactors of the ω2 and T 2 terms. The predic-tion from canonical Fermi liquid theory is that the fre-quency and temperature dependent scattering rate goesas

1/τ ∝ (~ω)2 + (pπkBT )2 (65)

where p = 226,27,126,127. It appears that in many casesthis simplest prediction has not been observed. Differentvalues have been observed: p ∼ 1 in URu2Si2

128, p ∼ 2.4in the organic material BEDT-TTF129, and p ∼ 1.5 inunderdoped HgBa2CuO4+δ

130. In contrast, the recentclaim for the very clean Fermi liquid Sr2RuO4 is thatscaling of ω and T are most consistent with p = 2131.However lower temperature and frequency data shouldbe taken. It has been proposed that the deviations fromp = 2 can be explained by elastic energy-dependent scat-tering, which can decrease the value of p below 227,126.Such a contribution depends on energy, but not on tem-perature and therefore contributes a ω2 but not a T2 termto the self-energy. See Ref. 27 for further discussion ofthese details and summary of the experimental situation.

Note that in Eqs. 58 and 59 there is the matter of howto define the plasma frequency ωp

15. For the electron

15 Note that this is only important for the overall scale factor ofthese expressions. Their functional dependence and form is in-dependent of the choice of the plasma frequency.

gas the plasma frequency is given by 4πne2/m, howeverthe precise definition is less clear in a real material withinteractions, interband transitions, and orbital mixing.In the above analysis ωp must come from the spectralweight of the full intra-band contribution. Rigorouslythe plasma frequency should be defined from the sumrule Eq. 54 as

∫ ∞0

σintra1 (ω) =ω2p

8. (66)

where σintra1 is contribution to the conductivity comingfrom all (and only) intraband spectral weight16 . Theintegral of this intraband contribution extends all theway to infinity, which gives of course a practical diffi-culty in real systems because the value of the spectralweight will be contaminated by interband contributions.Note that for a complicated interacting system the inte-gral and ωp must include not only the contributions ofthe weight of the low frequency Drude peak, but mustalso include higher energy parts of the spectra (satel-lites) if they derive from the same non-interacting band.It can be difficult experimentally to estimate ωp accu-rately. For instance, in the case of a system with strongelectron-phonon coupling the spectra may be approxi-mately modeled as the usual Drude model with a simpleω = 0 Lorenztian and frequency independent scatteringrate at low frequencies, but with a higher frequency satel-lite at the characteristic phonon frequency which resultsphysically from the excitation of a real phonon as well asan electron-hole pair. The high frequency satellite man-ifests in the scattering rate as an onset at the phononfrequency as an additional decay channel becomes avail-able at that energy. In this sense the plasma frequencywhich goes into an extended Drude analysis is the plasmafrequency associated with the spectral weight of both thelow frequency ‘Drude’ contribution and the satellite. Thespectral weight of the low frequency part may be clear,but it may be a difficult practical matter to clearly iden-tify the spectral weight of the satellite, due to the over-lapping contributions of it with true interband transitionsetc. In practice, a high frequency cutoff to the integral inEq. 66 is usually set at some value which is believed tocapture most of the intraband contribution, while mini-mizing contamination by interband transitions. A num-ber of different criteria can be used: the requirement thatthe integral is temperature independent, an appeal to re-liable band structure calculations etc. etc.

There can also be a difficulty in definitions of theplasma frequency for systems described by the Mott-Hubbard model, where a strong onsite repulsion splits

16 Note that in anything other than a simple Drude system theplasma frequency here is not given by the plasma minimum inthe reflectivity as that quantity is renormalized by a factor of

1√ε∞

. As mentioned above, this later quantity is known as the

screened plasma frequency or plasmon frequency.

33

a single metallic band into upper and lower Hubbard‘bands’. In such cases, one has one band or two depend-ing on definitions and so the matter of the true intrabandspectral weight can be poorly defined. In such cases it isimportant to remember that in this and other cases thatthe Drude model is a classical model applicable only toweakly interacting mobile particles. Although the EDMcan be put on more rigorous ground expressing its quan-tities as optical self-energies, one still is always makingan analogy and connection to a non-interacting system.To the extent that the EDM is just a parameterizationof optical spectra, it is always valid. However, to theextent that its output can be interpreted as a real massand scattering rate of something (i.e. a particle), it isimportant that that something exists. In other words, itmust be valid to discuss the existence of well-defined elec-tronic excitations in the Fermi-liquid sense in the energyregimes of interest.

In this regard, one must also be aware of the contri-bution of any interband terms in the evaluation of Eqs.58 and 59. These expressions for the EDM are only validfor excitations that are nominally ω = 0. In principlefinite frequency interband excitations (classically Drude-Lorentz) can overlap with a broad Drude intraband termgiving artifacts that should not be considered as contri-butions to the scattering rate and effective mass. Thesharp onset in 1/τ(ω) in Fig. 37 and the cusp in Fig. 38are due to a onset of strong interband excitations in thisspectral region (Fig. 19).

The EDM has been used in many different contextsin strongly correlated systems. An ideal example is af-forded by its application to heavy fermion systems. Asmentioned above, due to interaction of conduction elec-trons with localized moments the conduction electronscan undergo extremely large mass renormalizations ofa factor of a few hundred over the free electron mass.These mass renormalizations develop below a coherencetemperature T ∗. Such effects are reflected in the opticalspectra. As seen in Fig. 40, at high temperatures anEDM analysis of CeAl3 reveals a pure Drude-like, essen-tially frequency independent mass and scattering rate.The conventionally sized mass and large scattering rateat high temperatures reflect the effects of essentially nor-mal electrons scattering strongly and incoherently withthe localized moments17. The spectra show no partic-ularly interesting frequency dependence. Below the co-herence temperature at low frequencies significant renor-malizations to the mass and scattering rate are seen. Themass develops a significant enhancement and the scatter-ing rate drops. The coherence temperature T ∗ and thefrequency above which the unrenormalized values are re-

17 Note that the conventional EDM analysis expresses m∗ as a ratiowith the band mass mb if ωp is treated correctly as discussedabove. This is in contrast to other techniques like, for instance,de Haas- van Alphen that expresses m∗ as a ratio with the freeelectron mass me.

covered ωc are equivalent to each other to within factorsof order unity (with ~ = kB = 1). Remarkably the opti-cal mass as compared to the band mass is increased by afactor of almost 350. Note that although both the massand the low frequency scattering rate undergo dramaticchanges, the DC Drude conductivity is not changed sig-nificantly, as σDC ∝ τ/m and the respective changesapproximately cancel. This is reflected in the fact thatthe relation m∗(ω → 0)/mb is expected to be equivalentto τ/τ∗(ω → 0) to within factors of order unity22. Notethat these changes in scattering rate and mass are re-flected in the optical conductivity itself by the formationof a very sharp zero frequency mode (essentially a Drudepeak) which rides on top of the normal Drude peak22.

FIG. 40: Scattering rate and effective mass in CeAl3. (a)Frequency-dependent optical scattering rate for CeAl3 at fourtemperatures, compiled using both infrared reflectivity andmicrowave cavity data. (b) Frequency dependence of m∗ ob-tained from Eq. 59132.

More recent data from my group133 using THzspectroscopy on the Kondo-lattice antiferromagnetCeCu2Ge2 shows that a narrow Drude-like peak forms(Fig. 41 Left) at low temperatures, which is similar toones found in other heavy fermion compounds that donot exhibit magnetic order. Using this data in conjunc-tion with DC resistivity measurements, we obtained thefrequency dependence of the scattering rate and effectivemass through an extended Drude model analysis (Fig. 41Right). The zero frequency limit of this analysis yieldsevidence for large mass renormalization even in the mag-netic state, the scale of which agrees closely with thatobtained from thermodynamic measurements. This mass

34

FIG. 41: (Left) a-f) Real and imaginary parts of the complex conductivity of CeCu2Ge2 as a function of frequency for sixtemperatures in the frequency range of 0-1 THz. Solid lines indicate experimental data, while dashed lines show results of afit described in the text. The values of the DC conductivity are also shown as a solid circle. (Right) The renormalized massof CeCu2Ge2 as a function of frequency derived from the extended Drude model for three temperatures. Solid lines indicateexperimental data with dashed lines showing results of a fit described in the text. b) Renormalized mass of CeCu2Ge2 at zerofrequency as a function of temperature based on the zero frequency extrapolation of the extended Drude model fits. Note thelog scale on temperature axis. c) Scattering rate as a function of frequency with numerical fits shown in dashed lines for threetemperatures. d) Scattering rate at zero frequency as a function of temperature. The error bars indicate uncertainty, which isprimarily due to the ambiguity in determining the plasma frequency from FTIR measurements. From Ref. 133

renormalization in the low frequency and temperaturelimit is of order 50-100 times the band mass mb, whichis consistent with the enhancement inferred from specificheat measurements. In other systems, the expectation isthat non-Fermi liquid behavior may reveal itself in fre-quency dependent scattering with dependencies slowerthan quadratic. Please see Ref. 27 for a theoretical dis-cussion and Ref. 134 for a representative experiment onCeFe2Ge2.

Evidence that the renormalizations in the optical massreflect the existence of real heavy particles in these sys-tems and not just a convenient parametrization of theoptical spectral can be seen in the above discussed Fig.5 which shows a proportionality of the optically mea-sured mass to the linear coefficient of the specific heat(which is proportional to inverse of the near EF densityof states and hence the mass). It is remarkable that a pa-rameter determined dynamically - essentially by shakingcharge with an oscillating E-field - is precisely relatedto a quantity which is determined thermodynamicallyby quantifying the amount of heat absorbed. This graphprovides a remarkable demonstration of the quasiparticleconcept.

As mentioned, in addition to electron-electron scatter-ing, the effects of electron-boson interactions also revealthemselves in the optical spectra. Spectral features in theoptical conductivity deriving from electron-boson cou-pling were originally discussed in the context of Holsteinprocesses - the creation of a real phonon and an electron-hole pair with the absorption of a photon - in supercon-ducting Pb films117,125. The EDM analysis can reveal

significant information about the interaction of electronswith various bosonic modes. Allen derived an expressionfor the optical scattering rate at T=0K117

1

τ(ω)=

2π

ω

∫ ω

0

dΩ(ω − Ω)α2F (Ω). (67)

where α2F (Ω) is the phonon coupling constant multipliedby the phonon density of states.

In Fig. 42 the optical conductivity calculated for anelectron with impurity scattering coupled to a single Ein-stein phonon is shown for a few different temperatures.(See Ref. 135 for details regarding the calculation ofthe effects of electron-boson scattering in optical spec-tral.) The lower two panels show the scattering rate andmass enhancement generated from the optical conductiv-ity. At low temperatures the optical conductivity (Fig.42 (top)) reveals a sharp Drude peak at low frequencies,which results from the usual electron-hole excitations.The sharp onset in absorption at the phonon frequency isthe threshold for excitation of a real phonon. At thresh-old, electron-hole pairs are excited additionally, but it isthe phonon which carries away most of the energy. Thisis reflected in the scattering rate, which is flat and smallat low ω, but shows a cusp and sudden increase at ω0 asthe phase space for scattering increases discontinuously(Fig. 42 (middle)).

Although real phonons can only be excited abovethreshold, electron-phonon coupling manifests itself atenergies below threshold by an increase in the effectivemass (Fig. 42 (bottom)). Physically, this can be un-

35

FIG. 42: Electron-boson model calculations with a bosonspectral density A(Ω) = ω0δ(ω − ω0). The top panel givesthe optical conductivity and the lower two panels show thecorresponding scattering rate and mass enhancement. Thecoupling constant in this calculation is set to 1. From135.

derstood as a renormalization of the electron-hole pairenergy modified by the excitation of virtual phonons.This renormalization of the energy manifests itself as anincreased mass. The electron-hole pair can be viewed asdragging a cloud of virtual phonons with it. One can getan approximate measure of the boson coupling functionas136

α2F (ω) =1

2π

d2

dω2[ωRe

1

σ(ω)]. (68)

Extended Drude model analysis was applied to thescattering rate of the Drude transport of Bi2Se3 topo-logical insulator surface states in Ref. 137. A shown inFig. 43 an sudden increase of the scattering rate as afunction of frequency is seen. The frequency of scale ofthe increase is close to the scale of the previously ob-served Kohn anomaly of the surface β phonon, 0.75 THzat 2kF . When magnetic field was applied in the Drudepeak shifted to finite frequency in a fashion expectedfor a cyclotron resonance and it was observed that asit moved above this β phonon frequency it also broad-ened (Fig. 43). Both of these effects can be understoodfrom the perspective that when the excitation energy islarger than the energy of the phonon, a possible decaymechanism of the electron-hole pair is through excitingthe phonon.

Although the results of the calculation displayed in Fig.

FIG. 43: (Color) The scattering rate as a function of cyclotronfrequency (red). The fully renormalized scattering rate bymass through extended Drude analysis as a function of fre-quency (green). The black arrow indicates β surface phononfrequency at the Kohn anomaly. From Ref. 137.

42 is for phonons, the general idea holds for electron-boson coupling generically136,138 although, of course, thebosonic density of states is generally more complicatedthan a single sharp mode. For instance Tediosi et al.139

found a remarkable example of this concept in semi-metalbismuth, where the bosonic excitation is formed out ofthe collective excitations of the electronic ensemble it-self i.e. the plasmon. This is an effect that should bepresent in all metals, but is enhanced in a semi-metallike bismuth due to its very low carrier density and highε∞ ≈ 100. At low frequencies Tediosi et al.139 found intheir EDM analysis (Fig. 44) that the mass and scat-tering rate are approximately frequency independent asexpected for the Drude model. However, at higher fre-quencies there is sharp onset in the scattering rate at astrongly temperature dependent position. As one coolsbismuth, the charge density changes dramatically and theplasmon frequency (as given by the zero crossing of ε1)drops by a factor of almost two. It was found that theenergy scale of this sharp onset in scattering tracks theindependently measured plasmon frequency as shown inFig. 45. One can note the strong resemblance of thedata to the calculation of an electron interacting withsharp mode in Fig. 42. A strong coupling as such be-tween electrons and plasmonic electron degrees of free-dom may in fact be captured within the same HolsteinHamiltonian that is used to treat the electron - longitudi-nal phonon coupling and describe polarons. For this rea-son this collective composite excitation has been calleda plasmaron140,141 when exhibited in the single particlespectral function. The observation of Tediosi et al. wasthe first such observation optically.

36

400

300

200

100

0

1/!

(cm

-1)

8004000cm-1

1.2

1.0

0.8

0.6

0.4

0.2

0.0

m*/m

8004000cm-1

100500meV

100500meV

(a) (b)

290 K 230 K 170 K 100 K 20 K

FIG. 44: (Color) Frequency dependent scattering rate (a) andeffective mass (b), calculated from the optical data of ele-mental bismuth using ε∞. The low frequency scattering rateτ−1(ω) progressively falls as the temperature is lowered. Anapproximately frequency independent region is interrupted bya sharp onset in scattering and a decrease of the effectivemass, which was associated with plasmon scattering.

FIG. 45: (Color) A parametric plot of the sharp onset inscattering plotted versus the independently measured plas-mon frequency in elemental bismuth. The plasmon frequencychanges monotonically as the sample is cooled from room tem-perature to 15K.

EDM analysis has been applied extensively to the high-Tc cuprate superconductors20. In Fig. 46 I show theimaginary and real parts of the optical self-energy as de-fined in Eq. 61 above for a series of four doping levelsof the compound Bi2Sr2CaCu2O8+δ

142. The displayedsamples span the doping range from the underdoped tothe severely overdoped. Again one can notice the resem-blance of the data to that of the model calculation in Fig.42 and indeed data of this kind, along with similar signa-tures in angle-resolved photoemission was interpreted as

a generic signatures of electron-boson interaction in thecuprates.

FIG. 46: (Color) a-d, The frequency and temperature depen-dent optical scattering rate, 1/τ(ω) for four doping levels ofBi2Sr2CaCu2O8+δ. a, Tc = 67 K (underdoped); b, 96 K (op-timal); c, 82 K (overdoped); d, 60 K (overdoped). e-h, Thereal part of the optical self-energy, - 2Σop1 (ω) as defined in Eq.61. From Ref. 142.

The identity of this boson is a matter of intense cur-rent debate in the community. Some groups have pointedto a many-body electronic source138,142–144 related to the‘41 meV’ magnetic resonance mode discovered via neu-tron scattering145,146. Others have argued that these fea-ture’s presence above Tc, the relative doping indepen-dence of its energy scale, and its universality among ma-terial classes demonstrate a phononic origin and indicatesthe strong role that lattice effects have on the low-energyphysics147 18. We note that whatever the origin of this ef-

18 My (NPA’s) opinion is that these are by and large straw-manarguments and perhaps not the right way to approach the prob-lem of the origin of high-Tc. In a strongly correlated system likethe cuprates, it is clear that everything is strongly coupled to ev-erything else. Magnetism is strongly connected to lattice effectsand vice versa. In this regard, it is not surprising that experi-ments sensitive to the lattice show lattice effects and experiments

37

fect, it is almost certainly the same as that which causesthe upturn in the optical scattering rate20,138,142. Weshould also mention that much of the discussion regard-ing these self-energies and the like has implicitly assumedthat the only possibility of understanding such featuresare a consequence of some sort of electron-boson cou-pling, but this is not necessarily the case. There are pro-posals that purely electronic interactions and proximityto a Mott insulator can give such features148,149.

C. Frequency dependent scaling near quantumcritical points.

A quantum phase transition (QPT) is a zero-temperature transition between two distinct groundstates as a function of a non-thermal parameter, such asmagnetic field, pressure, charge density etc.150,151. Un-like conventional phase transitions, which are driven bythermal fluctuations, they are driven essentially by di-verging fluctuations of the system’s zero point motion.Once thought to be only of academic interest, the exis-tence of a QPT nearby in a material’s parameter space isbelieved to influence a whole host of finite temperatureproperties. QPTs may hold the key to understandingthe unusual behavior of many systems at the forefront ofcondensed matter physics150.

Just as in the case of conventional phase transitions,QPTs are believed to be characterized by diverginglength and time scales if second order. In the disorderedstate, one envisions an order parameter that fluctuatesslower and slower over longer and longer length scales asthe transition is approached until at the transition theselength and time scales diverge. These length and timescales are the correlation length ξc and time τc. Thedivergence of the fluctuation time scales, which in thecase of conventional transitions is called critical slowingdown implies the existence of a characteristic frequencyωc which vanishes at the transition.

In a QPT, the transition occurs as a function of somenon-thermal parameter K. It is usually considered thatclose to the transition at Kc, ξc diverges in a manner∝ |K−Kc|−ν and τc diverges ∝ |K−Kc|−zν ∼ ξz whereν and z are called the correlation length and dynamicexponent respectively.

From the classical case a well established formal-ism exists for the understanding of various physicalquantities close to continuous (2nd order) transitions.Widom’s scaling hypothesis assumes that close to thetransition the only relevant length and time scales arethose associated with diverging correlations in the orderparameter152. Such considerations hold for QPTs also

sensitive to magnetism show magnetic effects. One should notexpect in such materials that various degrees of freedom neatlypartition in separate distinct subsystems.

and means that physical observables such as magnetiza-tion or conductivity are expected to have scaling forms,in which independent variables such as the temperature,probe frequency or wave vector appear in the argumentonly as a product with quantities like ξc and τc. This isexpressed formally in terms of finite− size scaling. Foran incident wave vector q = 0 such a scaling form can bewritten

O(K,ω, T ) =1

T do/zF (ω/T, Tτc). (69)

Here O is some observable and F is the scaling func-tion. do is the scaling dimension, which determines howvarious physical quantities change under a renormaliza-tion group transformation. It is often close to the “en-gineering” dimension of the system (for instance 0 for2D conductivity), but can acquire an anomalous dimen-sion in certain circumstances. Generally one expects fre-quency and temperature dominated regimes of these re-sponse functions for ω T, ωc and T ω, ωc respec-tively. For further discussion on scaling forms and moti-vation on this and other related points see the excellentreview by Sondhi et al.150.

Scaling functions written as a function of ξc and τccan be very powerful as they allow the analysis of ex-perimental data independent of microscopic theory. Forinstance, the scaling exponents ν and z only depend oncertain global properties of the system such as symmetry,dimensionality, and the nature of the dominant interac-tions.

There are many studies where the existence of a QPTis inferred from frequency dependent scaling (see forexample153) in a regime where frequency effects domi-nate over temperature ones, but there are far fewer stud-ies where scaling is investigated as one passes througha known QPT. I know of only two where scaling in thefrequency dependent conductivity has been investigated.This is unfortunate because, as mentioned, such stud-ies could potentially be tremendously powerful for theinvestigation of correlated systems where the appropri-ate underlying physical model is not clear. This showsthe extreme experimental challenges inherent in accessingthe experimentally relevant frequency and temperatureranges where ~ω kBT while both ~ω and kBT are aslow as possible, but also below any higher energy scalesthat are not relevant to the ordered state. Hopefully thissituation will change with the increasing prominence andfeasiility of broadband microwave and THz techniques.

The first experimental study to investigate frequencydependent scaling near a QPT was that of Engel et al.154,who used a waveguide coplanar transmission setup toperform broadband microwave spectroscopy on quantumHall systems. In quantum Hall systems the longitudinalresistivity goes to zero at field values where the transverseresistivity assumes perfectly quantized values. Althoughthe value of the Hall resistivity is perfectly quantized, thewidth of the transition region depends on temperature155

38

and on measurement frequency154. In Fig. 47 the Reσxx vs. B is shown at three different frequencies andtwo different temperatures. One can see that at 50 mK,the width of the transitions regions (∆B) is strongly fre-quency dependent, while at 470 mK the widths are al-most frequency independent.

FIG. 47: Re σxx vs. B at three frequencies and two tem-peratures. Peaks are marked with Landau level N and spinindex.

FIG. 48: Peak width between extremal points in longitundinalresistance d Re σxx/dB ∆B vs. measurement frequency atthree different temperatures. Data are shown for the N = 0 ↓Landau level.

This frequency and temperature dependence of ∆B(as defined as the extrema of the derivatives of Re σxx)is summarized in Fig. 48 for one of the spin-split levels.

Similar data are found for other levels. One sees that forall such levels at low frequencies, the data are frequencyindependent and temperature dependent, whereas for lowtemperatures the data are temperature independent andfrequency dependent. The crossover between regimestakes place when 3~ω ∼ kBT . Such data are consis-tent with scaling theories of quantum criticality, wherethe only relevant frequency scale at the QPT is set by thetemperature itself. It is also significant that the crossovercondition between frequency and temperature dominatedregimes involves ~, showing that the physics is essentiallyquantum mechanical. In the high frequency data it isobserved that spin split levels give a ∆B which is ap-proximately ∝ ωγ , with γ = 0.43. This is consistentwith scaling theories that give an exponent for quantumpercolation of 1/γ = zν = 7/3 and also consistent withthe temperature dependence of these transition widths

as measured by DC transport that show a T1

7/3 depen-dence. It was claimed that through an analysis of boththe temperature and frequency dependent exponents, onecan extract the dynamic exponent z, which yields z = 1.This is consistent with long range Coulomb interaction.

Both the DC data155 and the high frequency data havetransition widths which exhibit a power law dependenceon temperature or frequency with an exponent of 3/7 atall spin-split transitions, showing that all such transitionsbetween quantum Hall levels fall into the same universal-ity class. It should be possible in such a study to incor-porate both temperature and frequency dependence intoa single scaling function although this has not been doneyet. It is expected that a general scaling function of theform

ρ(B, T, ω) = F (ω/T, δ/T1zν ) (70)

applies where δ = |B −Bc|/Bc measures the distance tothe quantum critical point. This function is equivalentto Eq. 69 using the fact that the scaling dimension of theresistivity vanishes in d = 2. In the high frequency andhigh temperature limits, Eq. 70 reduces to one where fre-quency or temperature appear in the argument as δ

ω1/zv

and δT 1/zv respectively.

The other frequency dependent study was that of Leeet al.156,157 who applied the ideas of finite size scalingto the 3D metal-insulator (MI) transition in Nb1−xSix.They measured the frequency dependent conductivity viamillimeter wave transmission through thin films whoserelative concentrations of Nb to Si were tuned exactly tothe MI QPT point at xc. As in the study of Engel etal.154, for samples tuned right to quantum criticality itis observed that for the ~ω kBT data the conductivityis temperature independent and that for ~ω kBT , thedata are frequency independent (Fig. 49). A crossoverfrom a frequency dominated regime to a temperaturedominated regime implies the existence of a scaling func-tion

∑of a form that can be inferred from above Eq.

69

39

FIG. 49: Reσ(ω) plotted against√ω for temperatures 2.8

to 32K and frequencies 87-1040 GHz. For the lowest tem-peratures and when ~ω > kBT , the data follow a

√ω. The

data approaches the DC values plotted on the vertical axisfor ~ω kBT .

σ(xc, T, ω) = CT1z

∑(~ωkBT

). (71)

In previous work on similar samples it was found thatfor samples tuned to the critical concentration the DCconductivity followed a T1/2 relation implying z = 2.As shown in Fig. 50, successful scaling can be found byplotting the scaled conductivity Re σ(T, ω)/CT 1/z us-ing this z. This procedure collapses the data over theentire measured frequency range for temperatures 16 Kand below, implying the applicability of a scaling func-tion that depends only on the scaled frequency. Datafor temperatures higher than 16 K starts to rise abovethe other collapsed curves, indicating the appearance ofother mechanisms (electron-phonon scattering for exam-ple) which limits the size of quantum fluctuations. Thisis consistent with the DC result where the conductivitystarts to deviate from the T

12 above 16 K.

As already inferred from Fig. 49, the scaling func-tion shows both temperature and frequency dominatedregimes, with a crossover at approximately ~ω ∼ kBT .Again, this shows that at the QPT the only energy scaleis set by the temperature itself. Note, that this crossoverhappens at a slightly different ω/T ratio than the quan-tum Hall case, but is still of order unity as expected.For high frequencies the scaling function follows a powerlaw dependence with the same exponent as was used toscale the vertical axis with z = 2. This means thatan equivalent scaling function could have been found bydividing the conductivity data by ω

12 and plotting the

data as a function of T/ω. Lee et al.156 attempted toguess the form of the scaling function with the expression∑

= Re(1 − ib~ω/kBT )1/2). The function successfully

FIG. 50: Log-log plot of conductivity scaled by the factorC = 475(ΩmK1/2)−1 vs. ω/T . For temperatures 16 K andbelow, the data for the entire experimental frequency rangecollapse onto a single curve within the experimental noise.Higher temperature data begin to rise above the collapseddata systematically for low scaled frequencies. The dashedline is a trial scaling function as described in the text.

captures the high and low ω/T , but misses the sharpcrossover near ~ω/kBT ∼ 1.

The value of the dynamic exponent z = 2, which wasfound in this study is interesting. It is different than whatis expected from models without a density of states sin-gularity, where z = d or for those with straight Coulombinteraction (z = 1 observed in the quantum Hall case).However, it is consistent with several field-theoretic sce-narios which give z = 2158.

I should mention that as presented above, frequencyscaling does not necessarily give us any information onexponents etc. that we could not also get from the tem-perature scaling. Here the primary importance of fre-quency scaling was in its ability to demonstrate howcharacteristic time scales diverge at the QPT, whichleaves the temperature itself as the only energy scalein the problem. However, as discussed by Damle andSachdev159, the fact that at the QPT itself, responsefunctions can written as a universal function of ω/T ,means that one does not necessarily expect the samebehavior in the ω = 0, T → 0 (incoherent) limit as inthe ω → 0, T = 0 (phase-coherent) limit . Since allDC experiments are in the former limit, while the vastmajority of theoretical predictions are in the latter, finitefrequency measurements can in fact given unique insight.It was predicted by Damle and Sachdev that at the 2D

40

FIG. 51: The real part of universal scaling function Σ, as afunction of ω = ω/T . There is a Drude-like peak from theinelastic scattering between thermally exciting carrier, whichfalls off at order ω ∼ 1. At larger ω there is a crossover to acollisionless regime. Importantly the function gives differentvalues in the ω → 0 and ω →∞ limits.

superfluid-insulator transition the conductivity is equalto two different universal numbers of order e2/h in theω/T → 0 and ω/T →∞ limits as shown in Fig. 51. Thisis true even if ω and T are both asymptotically small.

D. Using light to probe broken symmetries

Optical spectroscopies are most often used to probedynamical correlations in materials, but they are alsoa probe of symmetry. Polarization anisotropies are ofcourse sensitive to structural anisotropies, but have beenmuch less used as a probe of more exotic symmetrybreakings in ordered states. All physical properties ofa solid must respect its underlying crystal symmetries.Formally this is codified as Neumann’s principle thatstates that the symmetry transformations of any intrin-sic physical property of a crystal must include at leastthe symmetry transformations of the point group of thatcrystal160. This means that the possible discrete sym-metries of solid such as reflections, rotations, inversion,rotation-reflections and time-reversal symmetry may beprobed by selecting the incoming and outgoing polariza-tion states of light in either the linear response or non-linear regimes.

In linear response the transmission or reflection prop-erties of material can be captured by a 2 × 2 “Jones”matrix e.g. a second rank tensor. The symmetries ofa material typically manifest in the form of an algebrarelating matrix elements or overall constraints (transpo-sition, unitarity, hermiticity, normality, etc.) on the formof Jones matrix. This was worked out in gratuitous detailby me in Ref. 161. Jones matrices are particular usefulfor TDTS as one typically measures the complex trans-

mission function. In the most general case, the Jonesmatrix is comprised of 4 independent complex values. Inthe basis of x− y linear polarization it is

T =

[Txx TxyTyx Tyy

] [EixEiy

]=

[EtxEty

]. (72)

The above equation forms an eigenvalue-eigenvectorproblem. If light is sent through the system under testwith a well-defined linear polarization state in, for in-stance, the x direction, then the Faraday rotation (rota-tion of transmitted light) may be measured through therelation tan(θF ) = Txy/Txx.

Going forward, I will replace the entries Tij in Eq. 72with entries A,B,C,D for convenience e.g.

[Txx TxyTyx Tyy

]=

[A BC D

]. (73)

Consider the simplest examples coming from rotationand reflection symmetries. If a material possesses aRz(π/2) rotation symmetry (C4) around the z axis, itsJones matrix will be invariant under a π

2 rotation. Ap-plying the relevant rotation matrix to Eq. 73 gives

Rz(π/2) =

[0 1−1 0

],

˜T = R−1

z (π/2)T Rz(π/2) =

[D −C−B A

]. (74)

Hence the Jones matrix for a material with Rz(π/2) sym-metry must have the form

C4(z) =⇒ T =

[A B−B A

]. (75)

Other cases (and even time-reversal symmetry) canbe handled in a somewhat similar fashion as shownin Ref. 161. TDTS measurements probing and re-lated ones probing circular polarization anisotropies haveplayed an important role in determining symmetry in thecuprates4,5.

Even more information can be determined by probingthe non-linear optical response of materials. The non-linear response is determined by tensors of higher rankand hence are constrained by symmetries in an even moreexplicit fashion. For instance the second harmonic gener-ation (SHG) response is particularly sensitive to the pres-ence of global inversion symmetry because unlike the lin-ear electric-dipole susceptibility tensor, which is allowedin all crystal systems, the SHG electric dipole suscepti-bility vanishes in centrosymmetric point groups, leavingthe much weaker electric-quadrupole susceptibility as theprimary source of second harmonic signal. Measurementsof the nonlinear susceptibility have made important con-tributions in the detection of subtle broken symmetry

41

states of matter in cuprates and iridates6,7. Althoughsome aspects of these symmetries can – in principle –be measured through x-ray or neutron diffraction experi-ments displacements of oxygen or bond currents (as pro-posed for the pseudogap state of the cuprates) may notbe very sensitive to x-rays. Polarization experiments alsohave the unique advantage of being sensitive to true long-range order e.g. slow fluctuations of an order will not givea polarization anisotropy. This is unlike scattering styleexperiments that always have a finite frequency windowof the scattered particle and can in principle pick up asignal from a slowing fluctuating order.

FIG. 52: Angular anistotropy of the second harmonic gener-ation signal of YBa2Cu3Oy above and below the pseudogaptemperature T ∗. Polar plot of I2ω(ϕ) measured at T = 30 Kin Sin − Sout geometry from YBa2Cu3O6.92 (squares). Theblue curve is a best fit to the average of two 180 rotated do-mains with 2′/m point group symmetry as described in Ref.7. The fit to the T = 295 K data (pink shaded area) is overlaidfor comparison. From Ref. 7.

In Fig. 52, I show the data of Zhao et al. who used sec-ond harmonic generation to reveal the symmetries brokenat the pseudogap temperature in the cuprates7. Theyshow that spatial inversion and two-fold rotational sym-metries are broken at a temperature close to T∗ (Fig.53) while mirror symmetries perpendicular to the Cu-O plane are absent at all temperatures. This transitionoccurs over a wide doping range and persists inside thesuperconducting dome, with no detectable coupling toeither charge ordering or superconductivity. The resultssuggest that the pseudogap region coincides with an odd-parity order that does not arise from a competing Fermisurface instability and exhibits a quantum phase transi-tion inside the superconducting dome.

Other work has uncovered a odd-parity hidden orderin a spin-orbit coupled correlated iridate6 and a multi-

FIG. 53: Inversion symmetry breaking transition across thephase diagram of YBa2Cu3Oy. Temperature dependence ofthe normalized linear and SH response of YBa2Cu3Oy witha) y = 6.67, b) 6.75 c) 6.92 and d) 7.0 to their room tempera-ture values. Data were taken in Sin−Sout geometry at anglescorresponding to the smaller SHG lobe maxima at T = 295K. Blue curves overlaid on the data are guides to the eye. Theinversion symmetry breaking transition temperatures TΩ de-termined from the SHG are marked by the dashed lines. Thewidth of the shaded gray intervals represent the uncertaintyin TΩ. From Ref. 7.

polar nematic phase of matter in the metallic pyrochloreCd2Re2O7

8. This latter phase spontaneously breaks ro-tational symmetry while preserving translational invari-ance, but is odd under inversion symmetry. Such a mul-tipolar nematic order cannot be identified even in princi-ple using using conventional charge transport anisotropymeasurements because the loss of inversion symmetry ismanifested in the spin texture of the Fermi surface.

E. Pump-probe spectroscopy and non-equilibriumeffects

Thermal equilibrium is one of the cornerstones of mod-ern statistical mechanics and condensed-matter physics.By considering and averaging over ensembles that do notevolve in time, microscopic physical phenomena can berelated to macroscopic physical laws, in a general waythat is independent of the specific model used to describethem. Indeed, the idea of thermal equilibrium even al-lows us to define concepts as basic as that of tempera-ture itself. However to the increased ease of use of ultra-fast femtosecond lasers, non-equilibrium measurementsof solids have become of very prominent and really rep-

42

resent a wide open area of inquiry.The study of materials in a non-equilibrium configura-

tion present numerous challenges. Obviously if one takesa strongly correlated system (which is hard to under-stand by itself) and non-equilibrium effects (which arehad to understand by themselves) and combines them,simplicity is not the natural outcome. But there can bereasons for wanting to push physical systems out of equi-librium and study their behavior. At least three classesof phenomena and motivations for studying a condensed-matter system out of equilibrium come to mind.

1) Experiments are typically performed by pumpingthe system at one time and probing it at a later stage.After pumping, one hopes to learn something about therelaxation mechanisms and timescales of the equilibriumphase by watching the system relax and return to equi-librium through the decay of its elementary excitations.There is also the recent possibility of using multiplepulses of light to induce not just absorption but also emis-sion. This requires reasonably long lived excitations, butis the foundation for the incredibly powerful technique of2D spectroscopy that is used in in NMR (to excite nuclei)and in the optical range (to study atomic vibrations)162.The technique has been developed recently in the THzrange and applied to insulating spin systems9. I considerthis a tremendously promising area.

2) One may drive the system in such a way as to al-low access to material configurations (for example, timeaveraged structure or free charge density) that cannotbe accessed in equilibrium. Intense pump pulses maytherefore change the free-energy landscape and allow acompeting phase to be stabilized in a transient fashion.The hope is that the transient phase reflects equilibriumpossibilities in a larger generalized parameter space.

3) One may drive a system to achieve a non-equilibrium phase that cannot exist or is not stabilizedwithout a time - dependent driving field. In this sense,the time- dependent pump should be considered a termin the Hamiltonian. Recent work that claims that a light-illumination-driven Floquet topological insulator can bestabilized is of this variety163–165. There has also beena tremendous amount of interest in the possibility oflight-driven superconducting phases10,11 by the Cavallerigroup.

This has become a huge area of inquiry and aside fromthe above, I won’t go into more detail. Please see classic3

and recent reviews163,166–168 for further details.

F. Topological materials and Berry’s phase

Ordered states of matter have been traditionally cate-gorized by the symmetries they break. Upon the order-ing of spins in a ferromagnet or the freezing of a liquidinto a solid, the loss of symmetry distinguishes the or-dered state from the disordered one. However, in recentyears it was discovered that materials can also be distin-guished from each other by specific topological proper-

ties that are encoded in their quantum mechanical wavefunctions169–171. This discovery that topological proper-ties of quantum mechanical wave functions are importantto a wide class of quantum materials has heralded a rev-olution in condensed matter physics. Topological statesare now known to play a critical role in a wide rangeof condensed matter phenomena, from the well knownquantum Hall effect, 3D topological insulators (TIs), andWeyl semimetals (WSMs) to the fundamental propertiesof polarization and magnetization171.

In many cases, these topological properties are charac-terized by “topological invariants”, which are quantumnumbers sensitive only to topological properties and in-sensitive to system specific details. At the level of non-interacting electrons many of these topological proper-ties can be captured through the concept of the Berryphase incurred by an electron as it moves through theBrillouin zone in momentum (k) space171. The relatedquantities of the Berry curvature and Berry connectionact like a pseudo-magnetic field and pseudo-vector po-tential, respectively, in k-space. For instance, a 2D topo-logical invariant, the Chern number, is an integral of theBerry curvature over a 2D Brillouin zone and accountsfor the net Berry “flux” through the surface171. Thesetopological invariants underlie topological states, such astopologically protected surface states in topological insu-lators and bulk states in topological semimetals. The ex-otic phenomena exhibited by topological materials haveresulted in a host of studies aimed at elucidating theirproperties over the past decade. Furthermore, new ma-terials are continually being discovered, including type-II Weyl semimetals (which exhibit a tilted Weyl cone ink-space)169, nodal line semimetals (which have lines orrings in k-space over which bands touch), and quadraticband touching systems172–174. In the vast majority ofcases the studied materials can be understood in the con-text of weakly interacting fermions and do not properlyfall under the heading of strongly correlated electrons.Still, the fields have very much progressed together intel-lectually and there has been much cross-fertilization so Icover these issues here19.

In many topological materials – in at least some regime– the electron dynamics are expected to be semi-classical.For instance in topological insulator surface states inweak applied magnetic fields, the motion is captured bythe semi-classical expressions for cyclotron resonance137.See Fig. 43. But in other cases, there can be features di-rectly ascribable to novel aspects of the band structure.For instance, 3D Dirac and Weyl systems are predictedto have a number of interesting semi-classical transport

19 These materials collectively have been referred to as “QuantumMaterials”. As all materials are “quantum”, its not clear whatthis terminology means. That has not stopped many (myselfincluded) from using it. Perhaps the best we can do is fall backon Justice Potter Stewart’s famous dictum “I know it when i seeit”.

43

and optical effects that are diagnostic for this state ofmatter175,176. With small modifications, most of theseresults apply equally to Weyl and Dirac systems. In theabsence of impurities and interactions the free fermionresult for the conductivity in the low energy limit (wherequadratic or higher order terms in the dispersion can beneglected), which arises from interband transitions acrossthe Weyl or Dirac node when the chemical potential (EF )is at the Weyl or Dirac point is

σ1(ω) = Ne2

12h

|ω|vF, (76)

where vF is the Fermi velocity and N is number ofnodes175–178. This prediction is closely related to theprediction and observation in 2D Dirac system of singlelayer graphene that for interband transitions the optical

conductance should be G1(ω) = e2

~ , giving a frequencyindependent transmission that is quantized in terms ofthe fine structure constant α as T (ω) = 1 − πα179–181.In a 2D material like graphene the Kubo-Greenwood ex-pression for the conductance from interband transitionscan be written (for the chemical potential at the Dirac

point and T=0) as G1(ω) = πe2

ω |v(ω)|2D(ω) where v(ω)is the velocity matrix element between states with ener-gies±~ω/2 and g(ω) is the 2D joint density of states. Theuniversal conductance arises because the Fermi velocityfactors that come into the matrix element are canceledby their inverse dependence in the density of states. Inthese 3D Dirac systems, another factor of ω comes in inthe density of states yielding Eq. 76. Please see Fig. 54for results showing this linearity for the possible Diracsystem ZrTe5 and WSM TaAs182,183.

Note that Eq. 76 implies a logarithmic divergence ofthe real part of the dielectric constant through Kramers-Kronig considerations184,185. The corresponding imagi-nary conductivity is

σ2(ω) = − 2

πN

e2

12h

|ω|vF

log2ΛvFω

. (77)

where Λ is a UV momentum cutoff.Quite generally, in noninteracting electron systems

consisting of two symmetric bands that touch each otherat the Fermi energy the optical conductivity generi-cally has power-law frequency dependence with exponent(d−2)/z where d is the dimensionality of the system and zis the power-law of the band dispersion186. Such power-law behavior is a consequence of the scale-free natureof such systems. It has been argued187 that at a con-ventional continuous transition the optical conductivityshould scale as (d− z − 2)/z. Due to their scale free na-ture one may regard Dirac systems as intrinsically quan-tum critical with a dynamic exponent equal to the banddispersion power-law. With the usual substitution forthe effective dimensionality of a quantum critical systemdeff = d + z the generic power-law expression for the

FIG. 54: (Top) The optical conductivity of ZrTe5 at 8 K atfrequencies below 1200 cm−1. The red (dotted) line is thelinear fitting of σ1(ω). From Ref. 182. (Bottom) Opticalconductivity for TaAs at 5 K. The blue (steep) and black(shallow) solid lines through the data are linear guides to theeye. The blue (steep) line show the Weyl part of the spectrum,while the black (shallow) likely line comes from higher energynon-Weyl states. The inset shows the spectral weight as afunction of frequency at 5 K (red solid curve), which followsan ω2 behavior (blue dashed line). From183.

Dirac conductivity follows. Please see Ref. 169 for fur-ther details on the optical response of WSMs and DSMs.

Of course, even better than the above sort of experi-ments would be the ability to identify and measure re-sponse functions that directly reflect the topological in-variants of the system. There are two known cases wherethis has been done. The first is the classic observationof the quantized Hall conductance in 2D electron gasesunder a DC magnetic field, which directly reflects thetopological Chern number of the system188. The sec-ond example is the quantized magnetoelectric responseof a topological insulator, measured by my group usingTDTS189. This is a realization of the much celebratedaxion electrodynamics and quantized electromagnetic re-sponse proposed in the early days in the TI field190–192

and represents the second time in the annals of condensed

44

FIG. 55: Faraday rotation for a number of different Bi2Se3

samples of different thicknesses. Different thicknesses give dif-ferent charge densities at the surface. From Ref. 189. Dashedlines represent values for the universal values of the Faradayrotation. From Ref. 189.

matter physics when a topological invariant has beenmeasured in a solid state system by charge transport.As shown in Fig. 55, for fields higher than 5 T, we ob-served a quantized Faraday (and Kerr) rotations, but ina regime where the dc transport is still semiclassical as itgets shorted by the non-chiral side surfaces. A nontrivialBerry’s phase offset to these values gives evidence for ax-ion electrodynamics and the topological magnetoelectriceffect.

In both of the aforementioned examples where a topo-logical invariant has been measured, the response is ex-hibited in the linear response regime. One may askwhether other topological states of matter, such as Weylor Dirac semimetals (DSMs) or nodal line systems, alsohave quantized observables in response to electromag-netic fields, and/or if specific signatures of Berry phaseeffects in general can be deduced. In this regard, it hasbeen pointed out recently that it may be possible to ad-dress these questions by going beyond the linear opticalresponse into a regime where the material response de-pends nonlinearly on the incident electric field193. In-tense, ultrashort optical pulses (from THz to x-ray fre-quencies) provide a new way to not only probe thenovel phenomena displayed by topological matter, butalso to drive these materials into new regimes that wereinaccessible via conventional approaches. For the lat-ter, the advantages of nonlinear optical techniques havebeen demonstrated in a number of condensed matter sys-tems. For example, as discussed in Sec. V E in high-temperature superconductors, optical SHG has been usedto reveal the broken symmetries characterizing the pseu-dogap phase and intense mid-IR pulses have been usedto photoinduce superconductivity.

Recent theoretical studies have shown that the non-linear optical response of solids is intimately related totheir topological properties193. It was demonstrated us-

ing the Floquet formalism that various nonlinear opticaleffects, such as the shift current in non-centrosymmetricmaterials and the photovoltaic Hall response, can quitegenerally be described in a unified fashion by quantitiesinvolving the Berry connection and Berry curvature. Forinstance, it has been shown that the SHG signal can berelated to the shift vector R, which in turn can be di-rectly linked to the Berry connection193. In addition,under a DC magnetic (B) field, SHG is intrinsically sen-sitive to the chiral anomaly in WSMs, an effect uniqueto these materials that transfers particles between Weylpoints with opposite chiralities under parallel electric andmagnetic fields. Finally, the circular photogalvanic effect(CPGE) in WSMs is also sensitive to the Berry phase aswell as the chirality of Weyl fermions194.

One of the few nonlinear optical effects that has beenmeasured in these materials to date is SHG on 3DDirac materials. Theoretical studies indicate that SHGcan yield signatures of several quantum effects that inturn shed light on the underlying topology. For exam-ple, Ref. 207 demonstrates that transitions arising nearWeyl nodes between linearly dispersing bands in a WSMshould give a nearly universal prediction in the low fre-quency limit for the non-linear susceptibility (χ(2))

χ(2) =g(ω) < v2R >

2iω3ε0. (78)

With the density of states g(ω) proportional to ω2, theSHG signal is predicted to diverge as 1/ω. Although thisresult is reminiscent of the ω dependence of the opticalconductivity discussed above for these systems, the 1/ωdivergence is a unique signatures for inversion-breakingWSMs in particular because it vanishes in DSMs. How-ever, similar to the case of the linear in ω conductiv-ity, the SHG divergence will be cutoff by disorder andnonzero Fermi energy in real materials. Recently195 havefound a giant, anisotropic χ(2) at 800 nm (1.55 eV) inTaAs, TaP, and NbAs, which may be related to a highfrequency limit of this physics. In the spectral rangemeasured, the effect is of order 7000 pm/V , which is anorder magnitude larger than in GaAs, which is the ma-terial with the next largest coefficient. It is important toprobe SHG and the shift current at even lower frequencyto look for the dependence of ω. Recent important effortsin this direction have been made by Patankar et al.196,who show that the effect is even bigger at even lowerfrequency and who also derive a new theorem that re-lates the spectral weight of the nonlinear conductivity tothe third cumulant of the ground state, which is a quan-tity related to the skewness or intrinsic asymmetry ofthe ground-state polarization distribution. Other effectshave been proposed such as a nonlinear Hall effect arisingfrom an effective dipole moment of the Berry curvaturein momentum space197 and a photoinduced anomalousHall effect198 for WSMs and photogalvanic effects199 inDSMs.

In DSM systems there is no direct photocurrent with-

45

out driving electric field200, but because of their spin se-lective transitions the photoconductivity is anisotropicfor polarized radiation. Ref. 201 proposes that inversionsymmetry breaking WSMs with tilted Weyl cones (TypeII most effectively) and doped away from the Weyl point,will be efficient generators of photocurrent and can beused as low frequency IR detectors. Such a photocurrenthas recently been demonstrated via the circular photo-galvanic effect (CPGE)202 in TaAs. The CPGE is thepart of a photocurrent that switches its direction withchanges to the handedness of incident circular polariza-tion. It can be shown203 to be sensitive to the anoma-lous velocity derived by Karplus and Luttinger204 thatwas later interpreted as a Berry-phase effect205,206. 202also pointed out that such experiments can also measureuniquely the distribution of Weyl fermion chirality in theBZ.

An interesting proposal of Ref. 194 was that of a quan-tized response also in the CPGE of inversion symme-try broken WSMs that possess no mirror planes or four-fold improper rotation symmetries (e.g. structurally chi-ral). The CPGE usually depends on non-universal mate-rial details. Refs.194,207 predict that in Weyl semimetalsand three-dimensional Rashba materials without inver-sion and mirror symmetries, that the trace of the CPGEis quantized (modulo multi-band effects that were arguedto be small) in units of the fundamental physical con-stants. It is proposed that the currents obey the relation

1

2

[djdt− dj

dt

]= C

2πe3

h2cε0I (79)

where C is the integer-valued topological charge of Weylpoint and I is the applied intensity. Alternatively, theright hand side of the equation can be expressed asC 4παe

h I where α is the fine-structure constant. In thisexpression, the currents for left and right circular polar-ization are perpendicular to the polarization plane. Al-ternatively the quantity

[jsat − jsat

]may be measured if

the relaxation time τ is sufficiently long and known in-dependently. An attractive property of this response isthat it is related to the chiral charge on a single node.The total node chirality in the BZ must of course be zero,however this does not prevent a CPGE. In an P breakingmaterial with no mirror planes, the Weyl nodes of oppo-site chirality do not need to be at the same energy. Onenode can be Pauli blocked rendering it inert and givinga quantized response for some finite range in frequen-cies. The proposed double Weyl system SrSi2

208 thathas no mirror planes (unlike TaAs) or RhSi209 (which ispredicted to have six-fold-degenerate double spin-1 Weylnodes and a four-fold-degenerate node) may be good can-didates for this effect. Its predicted magnitude is wellwithin the range of current experiments. Using the uni-

versal coefficient e3

~2cε0= 22.2 A

W ·ps , 194 predict for τ ∼ 1

ps a steady state photocurrent of ∼ 2 nAW/cm2 , which is

approximately 100 times that found in the topologicalinsulator films210.

SHG can potentially also give insight into other ef-fects, such as the quantum nonlinear Hall effect, whicharises from an effective dipole moment of the Berry cur-vature in k-space211. Adding a B field makes SHG alsosensitive to the chiral magnetic effect (directly linked tothe chiral anomaly), the magnetochiral effect (a differ-ence in the material response to left and right circularlypolarized light (LCP and RCP, respectively), and theBerry curvature199,207. The relative strengths of theseresponses depend on the orientation of B with respectto the vector b in momentum space that connects theWeyl nodes, indicating that SHG measurements will alsoshed light on this important parameter. Second harmonicgeneration may be a relatively simple way to gain sub-stantial insight into the intrinsic topological properties of3D nodal semimetals.

Even more insight into topological invariants may begiven by high-order THz sideband generation (HSG),which has been proposed recently as a probe of the Berryphase in materials like MoS2 or bilayer graphene212,213.In this scheme a linearly polarized laser is used to excitecharge across the bandgap, after which elliptical or cir-cularly polarized radiation accelerates the charge arounda closed loop in the Brillouin zone. Upon recombination,the electron-hole pair emits polarized light, the plane ofwhich is predicted to be rotated by exactly the accu-mulated Berry’s phase in the loop213. One may mapout the momentum dependence of the Berry phase byusing successively higher THz pulse amplitudes. A re-lated experiment has been done recently by Banks etal.214 who find that a linear birifrigence in their high-order sideband generation (HSG) in semiconductors isaffected by Berry phases. Using a combination of a near-band gap laser beam, which excites a semiconductor thatis being driven simultaneously by sufficiently strong ter-ahertz (THz)-frequency electric fields, the highest-ordersidebands are associated with electron-hole pairs beingdriven coherently across roughly 10% of the Brillouinzone around the Γ point. The THz pulse adiabaticallydrives an electron coherently across a large portion ofthe Brillouin zone and picks up a Berry’s phase beforerecombination.

VI. ACKNOWLEDGMENTS

I would like to thank the organizers of the 2008 BoulderSchool for Condensed Matter Physics for the opportunityto first talk about these topics. These lecture notes werethen updated for the 2018 QS3 Summer School at CornellUniversity. I’d also like to thank Luke Bilbro, VladimirCvetkovic, Natalia Drichko, Amit Keren, and Wei Liufor helpful suggestions and careful reading of the originalversion of these notes.

46

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