Article
Reference
Electrodynamics of correlated electron materials
BASOV, D., et al.
BASOV, D., et al. Electrodynamics of correlated electron materials. Reviews of modern
physics, 2011, vol. 83, no. 2, p. 471-542
DOI : 10.1103/RevModPhys.83.471
Available at:
http://archive-ouverte.unige.ch/unige:23945
Disclaimer: layout of this document may differ from the published version.
1 / 1
Electrodynamics of correlated electron materials
D. N. Basov
Department of Physics, University of California San Diego, La Jolla, California 92093-0319,
USA
Richard D. Averitt
Department of Physics, Boston University, Boston, Massachusetts 02215, USA
Dirk van der Marel
De
´
partment de Physique de la Matie
`
re Condense
´
e, Universite
´
de Gene
`
ve,
CH-1211 Gene
`
ve 4, Switzerland
Martin Dressel
1. Physikalisches Institut, Universita
¨
t Stuttgart, Pfaffenwaldring 57, 70550 Stuttgart, Germany
Kristjan Haule
Department of Physics, Rutgers University, Piscataway, New Jersey 08854, USA
(Received 4 September 2009; published 2 June 2011)
Studies of the electromagnetic response of various classes of correlated electron materials including
transition-metal oxides, organic and molecular conductors, intermetallic compounds with d and
f electrons, as well as magnetic semiconductors are reviewed. Optical inquiry into correlations in all
these diverse systems is enabled by experimental access to the fundamental characteristics of an
ensemble of electrons including their self-energy and kinetic energy. Steady-state spectroscopy
carried out over a broad range of frequencies from microwaves to UV light and fast optics time-
resolved techniques provides complimentary prospectives on correlations. Because the theoretical
understanding of strong correlations is still evolving, the review is focused on the analysis of the
universal trends that are emerging out of a large body of experimental data augmented where
possible with insights from numerical studies.
DOI: 10.1103/RevModPhys.83.471 PACS numbers: 71.27.+a, 71.30.+h, 74.25.Gz, 78.20.e
CONTENTS
I. Introduction 472
II. Experimental Probes and Theoretical Background 474
A. Steady-state spectroscopy 474
B. Pump-probe spectroscopy 475
C. Theoretical background 477
D. Sum rules 478
E. Extended Drude formalism and infrared
response of a Fermi liquid 479
F. Dynamical mean field theory 480
III. Excitations and Collective Effects 482
A. Free charge carriers 482
B. Charge transfer and excitons 482
C. Polarons 483
D. Optical excitation of magnons 485
E. Power-law behavior of optical constants and
quantum criticality 486
F. Electron-boson interaction 488
G. Superconducting energy gap 489
H. Pseudogap and density waves 491
IV. Optical Probes of Insulator-to-Metal Transitions 492
A. Emergence of conducting state in correlated
insulators 492
B. Quasiparticles at the verge of localization 493
C. Superconductor-insulator transition 494
D. Conductivity scaling for metal-insulator transition 495
E. Photoinduced phase transitions 495
F. Electronic phase separation 497
G. Insights by numerical methods 498
V. Transition-metal Oxides 501
A. Cuprates 501
1. Steady-state spectroscopy 501
2. Pump-probe spectroscopy 502
B. Vanadium oxides 504
1. Steady-state spectroscopy 504
2. Pump-probe spectroscopy 505
C. Manganites 506
D. Ruthenates 509
E. Multiferroics 509
F. Iridates 510
G. Oxide heterostructures 511
VI. Intermetallic Compounds and Magnetic Semiconductors 512
A. Heavy-fermion metals 512
B. Kondo insulators 513
C. Beyond the Anderson model 515
D. Magnetic semiconductors 515
1. III-Mn-As 515
2. EuB
6
516
REVIEWS OF MODERN PHYSICS, VOLUME 83, APRIL–JUNE 2011
0034-6861= 2011=83(2 )=471(71) 471 Ó 2011 American Physical Society
3. Transition-metal silicides 517
E. Iron pnictides 518
VII. Organic and Molecular Conductors 519
A. One-dimensional molecular crystals 519
B. MX chains 520
1. Mott insulators 520
2. Peierls systems 521
C. Two-dimensional molecular crystals 522
1. Mott insulator versus Fermi liquid 522
2. Charge order and superconductivity 522
D. Graphene 523
VIII. OUTLOOK 525
I. INTRODUCTION
In their report on the Conference on the Conduction of
Electricity in Solids held in Bristol in July 1937, Peierls and
Mott wrote ‘‘Considerable surprise was expressed by several
speakers that in crystals such as NiO in which the d band of
the metal atoms were incomplete, the potential barriers be-
tween the atoms should be high enough to reduce the con-
ductivity by such an enormous factor as 10
10
’’ (Mott and
Peierls, 1937). The ‘‘surprise’’ was quite understandable. The
quantum mechanical description of electrons in solids— the
band theory, developed in the late 1920s (Bethe, 1928;
Sommerfeld, 1928; Bloch, 1929)—offered a straightforward
account for distinctions between insulators and metals.
Furthermore, the band theory has elucidated why interactions
between 10
23
cm
3
electrons in simpl e metals can be readily
neglected, thus validating inferences of free electron models.
According to the band theory NiO (along with many other
transition-metal oxides) are expected to be metals in conflict
with experimental findings. The term ‘‘Mott insulator’’ was
later coined to identify a class of solids violating the above
fundamental expectations of band theory. Peierls and Mott
continued their seminal 1937 report by stating that ‘‘a rather
drastic modification of the present electron theory of metals
would be necessary in order to take these facts into account’’
and proposed that such a modification must include Coulomb
interactions between the electrons. Arguably, it was this brief
paper that has launched systematic stud ies of interactions and
correlations of electrons in solids. Ever since, the quest to
fully understand correlated electrons has remained in the
vanguard of condensed matter physics. More recent inves-
tigations showed that strong interactions are not specific to
transition-metal oxides. A variety of d- and f-electron inter-
metallic compounds as well as a number of -electron or-
ganic conductors also revealed correlations. In this review we
attempt to analyze the rich physics of correlated electrons
probed by optical methods focusing on common attributes
revealed by diverse materials.
Central to the problem of strong correlations is an interplay
between the itineracy of electrons in solids originating from
wave function hybridization and localizing effects often
rooted in electron-electron repulsion (Millis, 2004).
Information on this interplay is encoded in experimental
observables registering the electron motion in solids under
the influence of the electric field. For that reason experimental
and theoretical studies of the electromagnetic response are
indispensable for the exploration of correlations. In Mott
insulators Coulomb repulsion dominates over all other pro-
cesses and blocks electron motion at low temperatures and
energies. This behavior is readily detected in optical spectra
revealing an energy gap in absorption. If a conduc ting state is
induced in a Mott insulator by changes of temperature and/or
doping, then optical experiments uncover stark departures
from conventional free electron behavior.
Of particular interest is the kinetic energy K of mobile
electrons that can be experimentally determined from the sum
rule analysis of optical data (see Sec. II.D) and theoretical ly
from band-structure cal culations. As a rule, experimental
results for itinerant electronic systems are in good agreement
with the band-structure findings leading to K
exp
=K
band
’ 1 in
simple metals (see Fig. 1). However, in correlated systems,
strong Coulomb interaction which has spin and orbital com-
ponents (Slater, 1929) impedes the motion of electrons,
leading to the breakdown of the simple single-particle picture
of transport. Thus, interactions compete with itinerancy of
electrons favoring their localization and specifically suppress
the K
exp
=K
band
value below unity (see Fig. 1). This latter
aspect of correlated systems appears to be quite generic and
in fact can be used as a working definition of correlated
electron materials. Correlation effects are believed to be at
the heart of many yet unsolved enigmas of contemporary
physics including high-T
c
superconductivity (see Sec. V.A.1),
the metal -insulator transition (see Sec. IV), electronic phase
separation (see Sec. IV.F), and quantum criticality (see
Sec. III.E).
Optical methods are emerging as a primary probe of
correlations. Apart from monitoring the kinetic energy, ex-
perimental studies of the electromagnetic response over a
broad energy range (see Sec. II.A) allow one to examine all
essential energy scales in solids associated with both elemen-
tary excitations and collective modes (see Sec. III).
Complementary to this are insights inferred from time-
domain measurements allowing one to directly investigate
FIG. 1 (color online). The ratio of the experimental kinetic energy
and the kinetic energy from band theory K
exp
=K
band
for various
classes of correlated metals and also for conventional metals. The
data points are offset in the vertical direction for clarity. From
Qazilbash, Hamlin et al., 2009.
472 Basov et al.: Electrodynamics of correlated electron materials
Rev. Mod. Phys., Vol. 83, No. 2, April–June 2011
dynamical properties of correlated matter (see Sec. IV). For
these reasons, optical studies have immensely advanced the
physics of some of the most fascinating many-body phe-
nomena in correlated electron systems.
More importantly, spectroscopic results provide an experi-
mental foundation for tests of theoretical models. The com-
plexity of the problem of correlated electrons poses
difficulties for the theoretical analysis of many of their prop-
erties. Significant progress was recently achieved by compu-
tational techniques including the dynamical mean-field
theory (DMFT) offering in many cases an accurat e perspec-
tive on the observed behavior (see Sec. II.F). The ability of
the DMFT formalism to produce characteristics that can be
directly compared to spectroscopic observables is particularly
relevant to the main topic of this review.
In Fig. 2 we schematically show possible approaches
toward an optical probe of interactions. It is instructive to
start this discussion with a reference to Fermi liquids (FL)
(left panels), where the role of interactions is reduced to mild
corrections of susceptibilities of the free electron gas (Mahan,
2000). The complex optical conductivity ~ð!Þ¼
1
ð!Þþ
i
2
ð!Þ of FL quasiparticles residing in a partially filled
parabolic band is adequately described by the Drude model
(see Sub. II.A for the definition of the complex conductivity).
The model prescribes the Lorentzian form of the real part of
the conductivity associated with the intraband processes
(Drude, 1900; Dressel and Gru
¨
ner, 2002; Dressel and
Scheffler, 2006):
~ð!Þ¼
N
eff
e
2
D
m
b
1
1 i!
D
¼
dc
1 i!
D
; (1)
where e is the electronic charge, N
eff
is the relevant density,
and m
b
is the band mass of the carriers which is generally
different from the free electron mas s m
e
, 1=
D
is the scatter-
ing rate, and
dc
is the dc conductivity. In dirty metals
impurities dominate and the scattering rate 1=
D
is indepen-
dent of frequency, thus obscuring the quadratic form of
1=ð!Þ that is expected for electron-electron scattering of a
Fermi liquid.
1
Nevertheless, this latter behavior of 1=ð!Þ has
been confirmed at least for two eleme ntal metals (Cr and
-Ce) through optical experiments (van der Eb et al., 2001;
Basov et al., 2002) using the so-called extended Drude
analysis (see Sec. II.A). Another characteristic feature of
Fermi liquid s in the context of infrared data is that the
relaxation rate of quasiparticles at finite energies is smaller
than their energy: 1=ð!Þ <!(at temperature T ! 0). The
contribution of interband transitions is also shown in Fig. 2
[right-hand-side peak in ð!Þ] and is usually adequately
described through band-structure calculations. The band-
structure results also accurately predict the electronic kinetic
energy of a Fermi liquid that is proportional to the area under
the intraband Drude contribution to the conductivity spectra
(see Sec. II.D).
One of the best understood examples of interactions is the
Eliashberg theory of the electron-boson coupling (Carbotte,
1990). Interactions with a bosonic mode at
0
modify the
dispersion of electronic states near the Fermi energy E
F
(top
panel in the middle row of Fig. 2). The spectra of 1=ð!Þ
reveal a threshold near
0
reflecting an enhancement of the
probability of scattering processes at !>
0
. The spectral
form of
1
ð!Þ is modified as well, revealing the development
of a ‘‘side band’’ in
1
ð!Þ at !>
0
. However, the total
spectral weight including the coherent Drude-like structure
and side bands is nearly unaltered compared to a noninteract-
ing system, and these small changes are usually neglected.
Thus, electron-boson interaction alone does not modify K
exp
with respect to K
band
. Importantly, characteristic features of
the bosonic spectrum can be extracted from the optical data
(Farnworth and Timusk, 1974). Various analysis protocols
employed for this extraction are reviewed in Sec. III.F.
Coupling to other excitations, including magnetic resonances,
also leads to the formation of sidebands that in a complex
system may form a broad incoherent background in
1
ð!Þ.
The right panels in Fig. 2 exemplify the characteristic
electronic dispersion and typical forms of the optical func-
tions for a correlated metal. Strong broadening of the disper-
sion away from E
F
indicates that the concept of weakly
damped Landau quasiparticles may not be applicable to
many correlated systems over the entire energy range. An
optical counterpart of the broadened dispersion are the large
Area =
n
eff
/ m
b
σ(ω)
ω
1/
τ
(ω
)
A(k,ω
)
1/τ(ω) = ω
ω
m*(ω)
ω
m
b
E
F
k
F
weak
correlations (LDA)
Area =
n
eff
/ m
b
ω
ω
ω
m
b
k
F
electron-boson
coupling
k
F
strong
correlations
Ω
D
Area =
n
eff
/ m
opt
ω
ω
1/τ(ω) = ω
ω
m
b
m
opt
Z
F
<1Z
F
<1
Z
F
=1Z
F
=1
ω
D
FIG. 2 (color online). Schematic diagram revealing complimen-
tary approaches to probing electronic correlations using IR and
optical methods. Top panels show the momentum-resolved spectral
function in a noninteracting metal (left), weakly interacting system
(middle), and strongly correlated system (right). Characteristic
forms of the real part of the conductivity
1
ð!Þ, the frequency-
dependent scattering rate 1=ð!Þ, and effective mass m
ð!Þ are
displayed. The Drude intraband contribution to the conductivity (the
low energy shaded area in the second row) develops a ‘‘sideband’’ in
a system with strong electron-boson coupling. The corresponding
enhancement of m
ð!Þ at energies below a characteristic bosonic
mode
0
can be registered through the extended Drude analysis
(see Sec. II.C). The magnitude of m
ð! ! 0Þ is related to the
quasiparticle renormalization amplitude Z introduced in Sec. II.C.
In a strongly correlated system (right panels) the oscillator strength
of the entire intraband contribution is suppressed with the spectral-
weight transfer to the energy scale of the order of U. The strength of
this effect can be quantified through the ratio of K
exp
=K
b
as in Fig. 1
or equivalently through the ratio of optical and band mass m
b
=m
opt
.
Quite commonly this renormalization effect and strong electron-
boson interaction act in concert yielding further enhancement of m
over the m
opt
at !<
0
.
1
See, for example, Abrikosov et al. (1963) , Pines and Nozie
`
res
(1966), and Ashcroft and Mermin (1976).
Basov et al.: Electrodynamics of correlated electron materials 473
Rev. Mod. Phys., Vol. 83, No. 2, April–June 2011
values of 1=ð!Þ. Finally, the low-energy spectral weight is
significantly reduced compared to band-structure expecta-
tions leading to K
exp
=K
band
that is substantially less than
unity. Suppression of the coherent Drude conductivity implies
the transfer of electroni c spectral weight to energies of the
order of intrasite Coulomb energy U and/or the energy scale
of interband transitions. These effects are routinely found in
doped Mott insulators, for example (see Sec. V ), as well as in
other classes of correlated materials.
2
It is instructive to discuss dynamical properties of corre-
lated electron systems in terms of the effective mass which in
general is a tensor m
k
. For a general dispersion
k
, the mass is
defined as m
1
k
¼ð1=ℏ
2
Þ@
2
k
=@k
2
, which reduces to a con-
stant for free electrons with a parabolic dispersion. Deviations
of m
k
from the free electron mass in simple metals are
adequately described by band-structure calculations yielding
m
b
. This quantity is frequency independent (bottom left
frame in Fig. 2) and enters the Drude equation for the com-
plex conductivity Eq. (1). Electron-boson interaction leads to
the enhancement of the effective mass compared to the band
mass m
b
at !<
0
as m
ð!Þ¼m
b
½1 þ ð!Þ, quantifying
the strength of the interaction (middle panel in the bottom
row). The frequency dependence of m
ð!Þ can be evaluated
from the effective Drude analysis of the optical constants.
Strong electron-electron interaction can radically alter the
entire dispersion so that m
opt
is significantly enhanced over
m
b
(right bottom panel). An equivalent statement is that K
exp
is reduced compared to K
band
(see also Fig. 1). Additionally,
electron-boson interactions may be operational in concert
with the corr elations in modifying the dispersion at
!<
0
. In this latter case one finds the behavior schemati-
cally sketched in the bottom right panel of Fig. 2 with the
thick line.
Because multiple interactions play equally prominent roles
in correlated systems, the resulting many-body state reveals a
delicate balance between localizing and delocalizing trends.
This balance can be easily disturbed by minute changes in the
chemical composition, temperature, applied pressure, and
electric and/or magnetic field. Thus, corr elated electron sys-
tems are prone to abrup t changes of properties under applied
stimuli and reveal a myriad of phase transitions (see Secs. III
and V). Quite commonly, it is energetically favorable for
correlated materials to form spatially nonuniform electronic
and/or magnetic states occurring on diverse length scales
from atomic to mesoscopic. Real space inhomogeneities are
difficult to investigate using optical techniques because of the
fairly coarse spatial resolution imposed by the diffraction
limit. Nevertheless, methods of near-field subdiffractional
optics are appropriate for the task (see Sec. V.B.1).
Our main objective in this review is to give a snapshot of
recent developments in the studies of electrodynamics of
correlated electron matter focusing primarily on works pub-
lished over the last decade. Introductory sections of this
article are followed by the discussion of excitations and
collective effects (Sec. III) and metal-insulator transition
physics (Sec. IV) exemplifying through optical properties
these essential aspects of correlated electron phenomena.
The second half of this review is arranged by specific classes
of correlated systems for the convenience of readers seeking a
brief representation of optical effects in a particular type of
correlated compounds. Given the abundant literature on the
subject, this review is bound to be incomplete in terms of both
topics covered and references cited. We conclude this account
by outlining unresolved issues.
II. EXPERIMENTAL PROBES AND THEORETICAL
BACKGROUND
A. Steady-state spectroscopy
Optical spectroscopy carried out in the frequency domain
from 1 meV to 10 eV has played a key role in establishing the
present physical picture of semiconductors and Fermi-liquid
metals (Dresse l and Gru
¨
ner, 2002; Burch et al., 2008) and
has immensely contributed to uncovering exotic properties of
correlated materials (Imada et al., 1998; Degiorgi, 1999;
Millis, 2004; Basov and Timusk, 2005). Spectroscopic mea-
surements in the frequency domain allow one to evaluate the
optical constants of materials that are introduced in the
context of materials parameters in Maxwell’s equations.
The optical conductivity is the linear response function relat -
ing the current j to the applied electric field E: jð!Þ¼
ð!ÞEð!Þ. Another commonly employed notation is that of
the complex dielectric function
~
ð!Þ¼
1
ð!Þþi
2
ð!Þ.
The real and imaginary parts of these two sets of optical
constants are related by
1
ð!Þ¼ð!=4Þ
2
ð!Þ and
2
ð!Þ¼ð!=4Þ½
1
ð!Þ1.
3
Absorption mechanisms
associated with various excitations and collective modes in
solids (Fig. 3) give rise to additive contributions to spectra of
1
ð!Þ and thus can be directly revealed through optical
experiments. In anisotropic materials the complex optical
constants acquire a tensor form. For instance, time reversal
symmetry breaking by an applied magnetic field introduces
nondiagonal components to these tensors implying interest-
ing polarization effects (Zvezdin and Kotov, 1997). In the vast
majority of optics literature it is assumed that the magnetic
permeability of a material ¼ 1 with the exception of
magnetic resonances usually occurring in microwave and
very far-infrared frequencies.
4
The complex optical constants can be inferred from one or
several complementary procedures (Dressel and Gru
¨
ner,
2002): (i) a combination of reflectance Rð!Þ and transmission
2
In transition-metal oxides the magnitude of the on-site Coulomb
repulsion can be both smaller or larger than the energy scale of
interband transitions (Zaanen et al., 1985). In organic conductors
the hierarchy of energy scales is consistent with a sketch in Fig. 2.
3
In general higher-energy contributions from interband transitions
b
ð!Þ (’’bound charge’’ polarizability) are present apart from the
quasifree electrons that are summarized in
1
replacing the factor 1
in this expression of
2
ð!Þ. The static bound charge polarizability is
defined as the zero-frequency limit of
b
ð!Þ, i.e.,
1
¼
b
ð0Þ.
4
This common assertion has recently been challenged by the
notion of ‘‘infrared and optical magnetism’’ (Yen et al., 2004;
Padilla et al., 2006; Shalaev, 2007) realized primarily in litho-
graphically prepared metamaterial structures but also in bulk co-
lossal magnetoresistance manganites (Pimenov et al. , 2005;
Pimenov et al., 2007). For inhomogeneous media, however, spatial
dispersion becomes relevant that in general mixes electric and
magnetic components (Agranovich and Ginzburg, 1984).
474 Basov et al.: Electrodynamics of correlated electron materials
Rev. Mod. Phys., Vol. 83, No. 2, April–June 2011
