Chemical pressure and hidden one-dimensional behavior in rare earth tri-telluride charge density wave compounds
ABSTRACT We report on the first optical measurements of the rare-earth tri-telluride
charge-density-wave systems. Our data, collected over an extremely broad
spectral range, allow us to observe both the Drude component and the
single-particle peak, ascribed to the contributions due to the free charge
carriers and to the charge-density-wave gap excitation, respectively. The data
analysis displays a diminishing impact of the charge-density-wave condensate on
the electronic properties with decreasing lattice constant across the
rare-earth series. We propose a possible mechanism describing this behavior and
we suggest the presence of a one-dimensional character in these two-dimensional
compounds. We also envisage that interactions and umklapp processes might play
a relevant role in the formation of the charge-density-wave state in these
[show abstract] [hide abstract]
ABSTRACT: In this lectures I discuss the electronic liquid crystal (ELC) phases in correlated electronic systems, what these phases are and in what context they arise. I will go over the strongest experimental evidence for these phases in a variety of systems: the two-dimensional electron gas in magnetic fields, the bilayer material Sr$_3$Ru$_2$O$_7$ (also in magnetic fields), and a set of phenomena in the cuprate superconductors (and more recently in the pnictide materials) that can be most simply understood in terms of ELC phases. Finally we will go over the theory of these phases, focusing on effective field theory descriptions and some of the known mechanisms that may give rise to these phases in specific models.04/2010;
arXiv:cond-mat/0606451v3 [cond-mat.str-el] 20 Jun 2006
Chemical pressure and hidden one-dimensional behavior in rare earth tri-telluride
charge density wave compounds
A. Sacchetti and L. Degiorgi
Laboratorium f¨ ur Festk¨ orperphysik, ETH - Z¨ urich, CH-8093 Z¨ urich, Switzerland.
DPMC, University of Geneva, 24, quai Ernest-Ansermet, CH1211 Geneva 4, Switzerland.
N. Ru and I.R. Fisher
Geballe Laboratory for Advanced Materials and Department of Applied Physics,
Stanford University, Stanford, California 94305-4045, USA.
(Dated: February 6, 2008)
We report on the first optical measurements of the rare-earth tri-telluride charge-density-wave
systems. Our data, collected over an extremely broad spectral range, allow us to observe both the
Drude component and the single-particle peak, ascribed to the contributions due to the free charge
carriers and to the charge-density-wave gap excitation, respectively. The data analysis displays
a diminishing impact of the charge-density-wave condensate on the electronic properties with de-
creasing lattice constant across the rare-earth series. We propose a possible mechanism describing
this behavior and we suggest the presence of a one-dimensional character in these two-dimensional
compounds. We also envisage that interactions and umklapp processes might play a relevant role
in the formation of the charge-density-wave state in these compounds.
PACS numbers: 71.45.Lr,78.20.-e
Low dimensionality is an important issue in solid
state physics , owing to the general tendency of low-
dimensional systems to form charge- and spin-density-
wave (CDW and SDW) states [2, 3]. CDW and SDW
phases are broken symmetry ground-states driven by
the electron-phonon and electron-electron interactions,
respectively. These phases, from the electronic point
of view, are induced by nesting of the Fermi surface
(FS) . Besides the single particle gap excitation, the
density-wave states are then characterized by the for-
mation of the collective CDW or SDW condensate .
Density-waves have been observed in several materials
such as linear-chain organic and inorganic compounds
. Strong interest in low-dimensional systems has also
been brought about by the considerable deviations of
their normal state properties from those of a Fermi liquid
[4, 5, 6, 7]. In one dimension, the Tomonaga-Luttinger-
liquid or Luther-Emery scenarios, implying phenomena
like spin-charge separation and non-universal power-law
behavior of the spectral functions, are most suitable .
In several cases, characteristic and peculiar power-law
behaviors were indeed observed in the spectroscopic (op-
tical and photoemission) properties of various quasi one-
dimensional materials [5, 6, 7]. It is worth noting that,
although the theory of density waves is well established
in the one dimensional (1D) case , little is known about
the two dimensional (2D) case.
Another class of compounds, which recently gained
importance in the study of density waves, are the rare
earth tri-tellurides RTe3 (R = La - Tm, excepting Eu
). These systems exhibit an incommensurate CDW,
stable across the available rare earth series . All these
compounds have the same average crystal structure (be-
longing to the Cmcm space group ) made up of square
planar Te sheets  and insulating corrugated RTe lay-
ers which act as charge reservoirs for the Te planes. The
lattice constant decreases on going from R = La to R =
Tm , i.e. by decreasing the ionic radius of the rare
earth atom. Therefore, the study of these compounds
allows to investigate the CDW state as a function of the
unit-cell volume and in particular of the in-plane lattice
constant a, which is directly related to the Te-Te distance
in the Te-layers.
Metallic conduction occurs along the Te-sheets leading
to highly anisotropic transport properties. For instance,
the ratio between the in and out of plane conductivity can
be as high as 3000 . The rare-earth atom is trivalent
for all members of the series  and thus the electronic
structure is quite similar in all of them. Band structure
calculations [9, 13, 14] reveal that the electronic bands
at the Fermi level derive from the Te pxand pyin-plane
orbitals, leading to a very simple FS, large part of which
is nested by a single wave-vector ? q = (0,x), x ≃ 0.29a∗
(a∗= 2π/a) in the base-plane of the reciprocal lattice
. This nesting appears to be the driving mechanism
for the CDW instability [15, 16].
The compounds are characterized by an unusually
large CDW gap, which was observed by angle re-
solved photoemission spectroscopy (ARPES) measure-
ments. The CDW gap ranges between 200 and 400 meV,
depending on the rare earth [15, 16, 17, 18, 19]. Con-
sistent with these large gap values, RTe3compounds are
well within the CDW state already at room temperature
and the CDW transition temperature (TCDW) is believed
to be even higher than the melting point . Owing
to the 2D character of these compounds, the gap is not
isotropic and shows a wave-vectordependence [16, 17]. In
particular, since the vector ? q does not nest the whole FS,
there are parts of it which are not gapped. Therefore, the
CDW state coexists with the metallic phase due to the
free charge-carriers in the ungapped regions of the FS.
The study of these compounds could give in principle an
important insight into the interplay between the metallic
state and the broken-symmetry CDW phase. Further-
more, the presence of a single nesting vector defines a
preferred crystallographic direction for the development
of the CDW state, and leads to features typical of a 1D
system, despite the 2D character of these compounds.
Besides to the prototype 1D systems , optical spec-
troscopy was already successfully applied to the study
of 2D chalcogenides such as NbSe2 and TaSe2, where
anomalous behavior of the carriers’ scattering rate was
observed in the CDW phase [20, 21], as well as NbSe3
and TaSe3, where a polaronic scenario for the CDW was
proposed [22, 23]. Here, we describe the first compre-
hensive optical study on RTe3. Optical spectroscopy is
in general an ideal tool to study CDW systems , since
it is able to reveal the opening of the CDW gap in the
charge excitation spectrum. The optical signature of the
CDW phase is in fact a finite-frequency peak, ascribed
to the transition from the CDW condensate to a single
particle (SP) state . Moreover, because of the imper-
fect nesting, as typical for quasi 2D systems like RTe3,
the optical technique also reveals the free-carriers contri-
bution in terms of a Drude peak.
II. EXPERIMENT AND RESULTS
We report on optical reflectivity measurements carried
out on RTe3 single crystals for representative members
across the rare earth series R = La, Ce, Nd, Sm, Gd,
Tb and Dy. Single crystal samples were grown by slow
cooling a binary melt, as described elsewhere . Plate-
like crystals up to several mm in diameter were removed
from the melt by decanting in a centrifuge. The crys-
tals could be readily cleaved between Te layers to reveal
clean surfaces for the reflectivity measurements. Exploit-
ing several spectrometers and interferometers, the optical
reflectivity R(ω) was measured for all samples from the
far-infrared (6 meV) up to the ultraviolet (12 eV) spec-
tral range, with light polarized parallel to the Te-planes.
Details pertaining to the experiments can be found else-
where [26, 27].
Figure 1 displays the overall R(ω) spectra for selected
members across the rare earth series. Consistently with
the large gap values, no temperature dependence of the
spectrum was observed between 2 K and 300 K. As ex-
pected from the presence of ungapped regions of the FS,
all samples exhibit a metallic R(ω), tending to total re-
flection at zero frequency (i.e., R(ω) → 1, for ω → 0),
and the appearance of a plasma edge around 10000 cm−1.
0 2000 4000 6000 8000
FIG. 1: R(ω) at room temperature of RTe3 (R=La, Ce, Nd,
Sm, Gd, Tb and Dy). The inset shows a blow up of the R(ω)
in the energy range between 0 and 10000 cm−1.
Above the plasma edge, several spectral features are also
observed.At lower frequency a bump is apparent in
R(ω) of all samples (see inset of Fig. 1). This feature is
more evident in LaTe3but it becomes progressively less
pronounced and shifts to lower frequency on going from
LaTe3to DyTe3, i.e., on decreasing the in-plane lattice
The large explored spectral range allows us to per-
form reliable Kramers-Kronig (KK) transformations. To
this end, R(ω) was extended towards zero frequency (i.e.,
ω → 0) with the Hagen-Rubens extrapolation (R(ω)=1−
2?ω/σdc) and with standard power-law extrapolations
at high frequencies . The DC conductivity values em-
ployed in the Hagen-Rubens extension of R(ω) are con-
sistent with transport measurements [25, 28]. The KK
transformations allow us to extract the real part σ1(ω)
of the optical conductivity, displayed in Fig. 2.
All σ1(ω) spectra in Fig. 2 are characterized by the
presence of two main features, namely a Drude peak,
revealing metallic conduction due to the free charge-
carriers, and a second mid-infrared peak centered at fi-
nite frequency (arrows in Fig. 2), corresponding to the
bump observed in R(ω) (Fig. 1). The depletion in the
σ1(ω) spectrum between these two features will be later
identified with the CDW gap. Higher energy excitations,
corresponding to the spectral features observed in R(ω)
above the plasma edge, are also present in σ1(not shown
in the figure) and they are ascribed to electronic inter-
band transitions. The frequencies of these excitations are
compatible with the predictions from band-structure cal-
culations [13, 14], taking into account the band energies
at the Γ-point. On the other hand, these band-structure
0 200040006000 8000
FIG. 2: Room temperature σ1(ω) of RTe3 (R=La, Ce, Nd,
Sm, Gd, Tb and Dy). Arrows mark the position of the SP
peak for each compound (see text). Inset: Drude-Lorentz fit
for DyTe3, showing the experimental data, the fitted curve,
the Drude and the Lorentz components.
calculations do not reveal any electronic transition below
1 eV for the undistorted structure (i.e., in the normal
state). It is thus natural to identify the depletion in the
σ1(ω) spectrum between the Drude and the mid-infrared
peaks with the CDW gap. Therefore, the mid-infrared
peak is ascribed to the charge excitation across the CDW
gap into a single particle (SP) state. In the following we
will refer to this peak as the SP peak.
Interestingly, our data show a clear red-shift of the SP
peak from LaTe3to DyTe3. This effect can be observed
directly from the σ1(ω) spectra (arrows in Fig. 2) or, to
a smaller extent, from R(ω) (inset of Fig. 1). In order
to get more quantitative information, a fit procedure,
exploiting the phenomenological Drude-Lorentz model,
was carried out on all spectra. It consists in reproducing
the dielectric function by the following expression:
˜ ǫ(ω) = ǫ1(ω) + iǫ2(ω) =
where ǫ∞is the optical dielectric constant, ωP and γD
are the plasma frequency and the width of the Drude
peak, whereas ωj, γj, and S2
quency, the width, and the mode strength for the j-th
Lorentz harmonic oscillator (h.o.), respectively. σ1(ω) is
then obtained from σ1(ω) = ωǫ2(ω)/4π.
The fit of all spectra were extended up to 40000 cm−1.
Besides the Drude contribution, five Lorentz h.o.’s for
each compound are required to fit the finite-frequency
features. This is explicitly shown in the inset of Fig. 2
for DyTe3, where the single fit components are displayed.
The three low frequency oscillators allow to reproduce the
rather broad absorption, ascribed to the SP peak. This
jare the center-peak fre-
choice is motivated by the fact that, for all samples, the
SP peak cannot be fitted with a single Lorentzian os-
cillator. There is indeed the presence of low- and high-
frequency shoulders, each described by a Lorentz h.o.,
which overlap to a background defined by a broad h.o. In
LaTe3and CeTe3the shoulder at the low frequency side
of the SP peak almost merges with the high frequency
tail of the Drude component. Therefore, the SP-peak in
each compound can be thought as composed by the su-
perposition of several excitations, which mimic the con-
tinuous distribution of gap values, as observed by ARPES
[15, 16, 17, 18, 19]. The remaining two high frequency
h.o.’s account for the optical (electronic interband) tran-
There are several interesting parameters, which can be
extracted from the fit. First of all, the plasma frequency
ωP, the square of which represents the total spectral
weight of the Drude peak. The larger ωP is, the higher
is the metallic degree of the system. It is also worth to
recall that ω2
P∝ n/m∗, where n is the charge-carriers’
density and m∗is the carriers’ effective mass. The ob-
tained values of ωP are plotted in Fig. 3a, as a function
of the lattice constant a . An increase of ωP going
from LaTe3to DyTe3is well evident.
As to the SP peak, since it is composed by three
Lorentz h.o.’s, we define the averaged quantity ωSP:
which represents the center of mass of the SP excitation
and thus provides an optical estimate for the CDW gap.
In Fig. 3b, ωSP is plotted as a function of a . The
decrease of ωSP (thus of the average gap value) is well
evident on going from LaTe3 to DyTe3, oppositely to
the observed increase of ωP (thus of the Drude spectral
weight, Fig. 3a). Another interesting quantity associated
to the SP-peak is the total spectral weight S2
(broad) SP excitation, which is defined as:
At this point we are then equipped for a thorough
discussion of our findings.
tail the evolution of the electronic properties across the
rare-earth series, we first focus our attention on the
relationship between the CDW and the normal state
(NS) properties for each compound. Although the high-
temperature NS cannot be studied experimentally since
TCDWis anticipated to be above the melting point of the
material, we can nevertheless draw some predictions by
exploiting our data and the band-structure calculations
[13, 14]. The Te px and py orbitals have a strong 2D
character (i.e., they have a negligible dispersion along the
direction orthogonal to the Te layers). These bands are
well approximated by a tight-binding model  in which,
taking into account the stoichiometry, the Fermi level
Before addressing in de-
ωP (×103 cm-1)
ωSP (×103 cm-1)
lattice constant (Å)
FIG. 3: Plasma frequency (a) and single-particle peak fre-
quency (b) as a function of the in-plane lattice constant .
The rare-earth atom for each compound is shown in the plot.
The solid lines are guides to eyes.
lies above half-filling (i.e., at 3.25 eV below the top of
the bands) , indicating that charge carriers are hole-
like. Considering a parabolic expansion of the 2D bands
around their maximum, the Fermi energy EF = 3.25 eV,
within a 2D free-holes model, is then related to the effec-
tive mass mNSby:
where n2Dis the 2D hole density in NS. This latter quan-
tity is determined by assuming 1.5 holes for each tel-
lurium atom within the Te layers and 2 Te atoms for
each square unit within the 2D layers (i.e., n2D= 3/a2).
From n2Dand EF, we achieve mNS for each sample by
exploiting eq. (4). The mNSvalues are reported in Table
The mNS values allow us to check our data in a self-
consistent manner. If we assume the conservation of the
spectral weight between the CDW and the normal state,
and that there will be no SP peak in the (hypothetical)
NS, the Drude contribution in NS would then have a
total spectral weight given by S2
Φ = ω2
TABLE I: Effective mass mNS and carriers’ density nNS in
the normal state and effective mass mCDW and carriers’ den-
sity nCDW in the CDW state for all samples. Carriers’ den-
sities and effective masses are given in carriers/cell and me
units, respectively. The last column reports the ratio Φ be-
tween the Drude spectral weights (see text) in the CDW and
normal state. Experimental uncertainties in the last digits
are given in brackets.
Therefore, from S2
three-dimensional carriers density nNSin NS, as reported
in Table I for each compound.
compared with the carriers’ density nch obtained from
the chemical counting. Since the 3D unit cell contains
four Te layers (i.e., 8 Te atoms) with 1.5 holes for each
Te atom, we get nch= 12 holes/cell. The fair agreement
(at least within the experimental uncertainties) between
nNS and nch is well evident in Table I.
makes us confident about the reliability of our analysis.
We now consider the free charge carriers surviving in
the CDW state. We can estimate their effective mass
mCDW from the existing specific heat data on LaTe3.
The Sommerfeld γ-value of the linear term in the specific
heat is γ = 0.0011 J mol−1K−2for LaTe3[25, 29]. We
assume a 2D free-holes scenario for the ungapped carriers
also in the CDW phase. Within this approach, γ and
mCDW are related by:
NS∼ nNS/mNS we can estimate the
This quantity can be
Since γ is difficult to extract from heat capacity mea-
surements for the magnetic members of the rare earth
series, we use the value of LaTe3for all samples. Bear-
ing in mind the small chemical and structural changes
occurring across the rare-earth series, this assumption
appears reasonable. Therefore, we can estimate mCDW
from eq. (5) for all samples, as reported in Table I.
It turns out that the free-carriers’ mass decreases by
a factor 2 on entering the CDW state.
can be explained by noting again that in RTe3 the rel-
evant electronic states correspond to the two orthogo-
nal bands deriving from the Te pxand pyorbitals. The
band structure, while nearly 2D, is anisotropic in the
plane. Therefore, mNS and mCDW should be consid-
ered as parameters describing the average curvature of
the free charge carriers bands in the normal and CDW
state, respectively. In this respect, our estimates of
the effective masses represent a harmonic average of the
mass per carrier, i.e., 1/mNS = 1/nNS
1/mCDW = 1/nCDW
over the ungapped states in the normal and CDW phases,
respectively. The difference in the mNSand mCDW val-
ues thus indicates that in the CDW phase the average
band curvature is larger than in NS. Consequently, the
CDW mainly gaps “heavy” carriers (i.e. those belong-
ing to states with small band-curvature). The remain-
ing ungapped carriers thus occupy states with a larger
band-curvature, resulting in smaller effective mass. Since
states with small band-curvature are flatter (i.e. they
have a smaller dispersion) than those with large band-
curvature, one can expect the former to give a larger con-
tribution to the density of states than the latter. There-
fore, it is not at all surprising that the CDW tends to gap
states with the small band-curvature. This way there are
more states which can be gapped and the CDW energy-
gain is larger.
The knowledge of mCDW and ωP allows us to deter-
mine the density nCDW of the ungapped carriers in the
CDW phase. The nCDW values for each sample are re-
ported in Table I. The large difference between nNSand
nCDW and consequently the dramatic reduction of the
number of ungapped charge carriers indicate the strong
effect of the CDW formation; a large part of the charge
carriers takes part in the formation of the CDW conden-
sate. That a significant part of the FS is gapped in the
CDW phase, is confirmed by ARPES data from which
the fraction of the ungapped FS can be estimated to be
10-20% [15, 16, 17, 18].
In this respect, the quantity Φ = ω2
interest, since it represents the ratio between the spectral
weight of the Drude peak and the total spectral weight
of the Drude term and the SP peak (Table I). Φ roughly
measures, as previously shown in Ref. 22, the fraction of
the ungapped FS area (i.e., those parts of FS which are
not affected by the CDW state). Since the effective mass
of the free carriers is different in CDW and NS, Φ does
not simply reduce to nCDW/nNSand it is larger than the
latter quantity by a factor given by mNS/mCDW ∼ 2.
Although the Φ values seem to significantly underesti-
mate the fraction of the ungapped FS compared to the
estimate evinced from ARPES data [15, 16, 17, 18], they
at least qualitatively confirm that a large portion of FS
is gapped. The difference between ARPES and optics, as
far as the gapping of FS is concerned, may also be recon-
ciled in part by taking into account the energy resolution
of the ARPES data . We now turn to the evolution
of the CDW state across the rare-earth series. First of
all, the reduction of ωSP on decreasing a (Fig. 3b) may
be considered as an indication for the diminishing impact
of the CDW state, when going from La to Dy. Such a
gap reduction is also consistent with the ARPES data
, showing a gap of about 400 meV in CeTe3[15, 16]
and of 200 meV in SmTe3. In recent ARPES exper-
iments  a reduction of the maximum gap-value with
decreasing a was also observed for several compounds of
i1/mi, where the two sums run
SP) is of
Φ = ωP
lattice constant (Å)
FIG. 4: Ratios Φ = ω2
of the ungapped FS) and nCDW/nCDW as a function of the
in-plane lattice constant . The left and right scales differ
by a factor of 2. The rare-earth atom for each compound is
shown in the plot. The solid line is a guide to eyes.
SP) (indicating the fraction
the RTe3series. It is important to remark that, differ-
ently from ARPES where the maximum gap-value can
be determined, our optical estimate provides a sort of
averaged value for the CDW gap over the whole FS.
In Fig. 4, Φ (Table I) is plotted as a function of the
lattice constant a . The portion of the ungapped FS
increases on decreasing a and has a more gradual and
less scattered behavior than ωP (Fig. 3a). Since Φ cor-
responds to the ratio between spectral weights, it is less
affected by possible uncertainties of the σ1(ω) absolute
value. In the same figure, the ratio nCDW/nNSbetween
the carrier’s densities in the CDW and NS (see Table I)
is also plotted as a function of a. As discussed above,
the two quantities have the same dependence on a and
just differ in absolute value by a factor ∼ 2 (Fig. 4),
coming from the effective mass ratio. The increase of Φ
with decreasing a is quite abrupt for rare earths beyond
the Sm-containing compound, indicating the onset of a
chemical pressure effect for a ≤ 4.34˚ A. On the other
hand, the effective mass is almost constant across the
rare-earth series so that the band-shape and the inertia
of the free charge carriers are only subtly affected by the
lattice compression. Therefore, this seems to rule out a
large change in the band width. Nevertheless, one cannot
exclude a priori that a yet subtle narrowing of the bands
with increasing lattice constant could lead to a better
nesting and to differences in the amount of the nested
In this context, it is then quite natural to assume that
the reduction of ωSP (Fig. 3b) on going from LaTe3to
DyTe3could be ascribed to a suppression of the nesting
condition, due to the changes in FS, because of the lat-
tice compression. In this scenario, the increase of the un-
gapped portions of FS with decreasing a (Fig. 4) leads to
an enhanced optical contribution due to the free charge
456 7 8 9 10
FIG. 5: σ1(ω) of RTe3 (R=La, Ce, Nd, Sm, Gd, Tb and
Dy) plotted on a bi-logarithmic scale. The y axis logarithmic
scale is vertically shifted for the sake of clarity. The scale is
the same for all samples and the scale-offset is shown for each
spectrum. The solid lines are power-law fits to the data (the
exponents are given in the figure).
carries. The strong decrease of ωSP is consistent with
the reduction of the perfectly-nested regions (where the
CDW gap is close to its maximum value) in favor of the
non-perfectly-nested regions (where the gap is close to
Finally, a power-law (σ1(ω) ∼ ωη) behavior for the
high frequency tail of the SP peak is observed in all sam-
ples. This is shown in Fig. 5, where σ1(ω) is plotted on bi-
logarithmic scale. The power-law behavior extends over
a rather limited energy interval, which at most is one or-
der of magnitude wide for the Dy compound. Power law
scaling is indicative of quasi 1D behavior [31, 32]. The
observed exponents, although not precisely determined,
give indications as to the mechanism behind CDW for-
mation in these materials. In the following analysis we
make the case that electron-electron interactions play a
crucial role in the CDW formation in RTe3.
The traditionally invoked mechanism for CDW forma-
tion is the electron-phonon coupling. In that case there
are three energy scales to be considered for optical ab-
sorption: the typical phonon frequency ω0, the single
particle (or Peierls) gap ∆, and the frequency ω at which
the measurement is done. As anticipated above, the ad-
vocated FS of the material  consists of two sheets of
open FS’s of a quasi 1D material (associated to the pxand
py orbitals, respectively). The measured vector for the
CDW modulation is very close to the vector that corre-
sponds to the nesting of the two sides of this quasi 1D FS.
Attributing the one dimensionality to the fact that RTe3
have a nearly perfect nested quasi-1D FS is of particular
relevance here, since the charge transfer integral (tperp)
along the transverse direction (i.e., describing the hop-
ping between the pxand pyorbitals) is not small and is
much larger than the temperature of the measurements.
Indeed, tperp> T would normally lead to coherent trans-
verse hopping, so that FS would have significant warping
in the transverse direction and the material would not
be 1D anymore. The warping of FS would loose its rele-
vance only at ω > tperp. However, this is not the appro-
priate situation for RTe3, since tperp= 0.37 eV [13, 15],
while the power law behavior is observed for frequencies
ω > 0.2 eV. But if nesting is strong and occurs with a
well defined ? q vector, then the system still acts as a 1D
system would essentially do. The 1D character, indicated
by the high frequency power law behavior of σ1(ω) (Fig.
5), may then persist even for ω ? tperp, provided that one
looks at phenomena involving the nesting wave-vector.
In the case of a 1D material, one would get different ex-
ponents for the optical behavior above the Peierls gap de-
pending on the hierarchy of the energy scales, pointed out
above. If one is in the so-called adiabatic limit ω0≪ ∆
where the phonon frequency is quite small, then one can
assimilate the potential created by the phonons to ei-
ther a periodic (with wave-vector ? q) static deformation,
if ω ≪ ω0, or to a quenched disorder varying in space,
if ω ≫ ω0. In the first case the conductivity can easily
be computed by looking at the scattering on the static
periodic potential with a wave-vector ? q, the CDW modu-
lation vector, using the techniques explained in Ref. 33.
One finds that σ1(ω) ∝ ωKρ−4where Kρ < 1 is the
Luttinger liquid parameter. Kρ = 1 for a Fermi liquid
or if interactions are weak and decreases with increas-
ingly repulsive interactions. It is clear that such a result
would not be compatible with the data, making it hard
to have the standard adiabatic mechanism for the CDW
formation. The other case, when the phonon potential is
viewed by the electron as a quenched disorder, is also not
quite compatible with the data. Indeed in that case one
finds σ1(ω) ∝ ωKρ−3. A weakly interacting system
would thus give exponents of order −2 or slightly below,
in somewhat closer but yet very unsatisfactory agreement
with the data. It thus seems unlikely that the data are
explained by the standard scattering over a distortion due
to adiabatic phonons. The data would on the contrary be
in reasonable agreement with the so-called antiadiabatic
limit (∆,ω) ≪ ω0, in which one can integrate over the
phonon field. The phonon fluctuations introduce then an
effective interaction between the particles. This interac-
tion corresponds to an umklapp scattering, which gives
σ1(ω) ∝ ω4Kρ−5. (6)
This would lead to exponents η close or slightly smaller
than −1, in much better agreement with the data. It is
however extremely unlikely to be able to find phonons
of such high frequency, since one would need ω0 > 1
eV. Consequently, phonons alone could not explain the
observed optical data and power laws.
A much more probable source for such an umklapp
scattering leading to eq. (6) is the direct interaction be-
tween electrons. Such an interaction can directly produce
an umklapp scattering, allowing the transfer of two par-
ticles from one sheet of the FS to the other and thus
transferring 4? q to the lattice . Note that the advo-
cated wave-vector  for the CDW modulation is in-
deed giving a value for 4? q close to a reciprocal lattice
vector, and thus allowing such umklapp processes to be
effective. An umklapp process leading to a power law
behavior in σ1(ω) and in other response functions was
theoretically predicted for strict 1D systems within the
Tomonaga-Luttinger-liquid scenario  and observed ex-
perimentally in the 1D Bechgaard salts with η = −1.3
[7, 31, 32]. Furthermore, we identify in σ1(ω) of the Ce
and La compounds a low-frequency tail of the SP peak
following a power-law behavior as well (Fig. 5). The re-
sulting exponent ranges between 2.2 and 3, which again
compares quite well with the value of 3 predicted for a
1D Mott insulator for which umklapp is the dominant
source of scattering .
Even though the exponents for the high-frequency
power-law observed in our data do not show a defined
trend across the rare-earth series, their values range be-
tween -1.7 and -1.0. This is in decent agreement with
such a scenario where interactions and umklapp would be
responsible for the observed behavior. This strongly sug-
gests that interactions rather than a standard electron-
phonon mechanism could play, in the rare-earth tri-
tellurides, a crucial role in the CDW formation as well.
Of course more studies both theoretically and experimen-
tally would be useful to ascertain the respective roles of
the interactions and of electron-phonon coupling with re-
spect to the CDW formation. Theoretically, a more care-
ful treatment of the effects of the transverse warping of
FS would be clearly needed, not only for the high energy
behavior but even more for the low frequency part of the
optical conductivity, much below the single particle peak.
In summary, we reported on the first optical measure-
ments of seven different rare-earth tri-tellurides.
data allow for a detailed analysis of both the Drude con-
tribution, ascribed to the free charge carriers resulting
from the presence of ungapped regions of FS, and the SP
peak, due to carriers’ excitation across the CDW gap.
On decreasing the lattice constant, a slight enhancement
of the metallic contribution and a simultaneous reduc-
tion of the CDW gap are observed. We propose that this
effect might be due to a quenching of the nesting condi-
tion driven by a modification of FS because of the lat-
tice compression. We also observe power-law behaviors
in σ1(ω), typical of a Tomonaga-Luttinger-liquid system,
which emphasize a non negligible contribution of 1D cor-
relation effects in the physics of these 2D compounds.
This also anticipates that interactions and umklapp pro-
cesses could play a significant role in the CDW formation
in these compounds.
The authors wish to thank J. M¨ uller for technical help,
and V. Brouet and M. Lavagnini for fruitful discussions.
This work has been supported by the Swiss National
Foundation for the Scientific Research within the NCCR
MaNEP pool. This work is also supported by the Depart-
ment of Energy, Office of Basic Energy Sciences under
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