# Angle-resolved photoemission study of the evolution of band structure and charge density wave properties in RTe_ {3}(R= Y, La, Ce, Sm, Gd, Tb, and Dy)

**ABSTRACT** We present a detailed angle-resolved photoemission spectroscopy (ARPES) investigation of the RTe3 family, which sets this system as an ideal “textbook” example for the formation of a nesting driven charge density wave (CDW). This family indeed exhibits the full range of phenomena that can be associated to CDW instabilities, from the opening of large gaps on the best nested parts of Fermi surface (up to 0.4 eV), to the existence of residual metallic pockets. ARPES is the best suited technique to characterize these features, thanks to its unique ability to resolve the electronic structure in k space. An additional advantage of RTe3 is that the band structure can be very accurately described by a simple two dimensional tight-binding (TB) model, which allows one to understand and easily reproduce many characteristics of the CDW. In this paper, we first establish the main features of the electronic structure by comparing our ARPES measurements with the linear muffin-tin orbital band calculations. We use this to define the validity and limits of the TB model. We then present a complete description of the CDW properties and of their strong evolution as a function of R. Using simple models, we are able to reproduce perfectly the evolution of gaps in k space, the evolution of the CDW wave vector with R, and the shape of the residual metallic pockets. Finally, we give an estimation of the CDW interaction parameters and find that the change in the electronic density of states n(EF), due to lattice expansion when different R ions are inserted, has the correct order of magnitude to explain the evolution of the CDW properties.

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- Tzong-Ru T. Han, Zhensheng Tao, Subhendra D. Mahanti, Kiseok Chang, Chong-Yu Ruan, Christos D. Malliakas, Mercouri G. Kanatzidis[Show abstract] [Hide abstract]

**ABSTRACT:**The electron-phonon mechanism that gives rise to various charge-ordered systems is often controversial because of the cooperative nature of the transformation, and because the structural aspect of the transformation is generally poorly understood. Using femtosecond electron crystallography, we reveal a two-step (≈400 fs and 3.3 ps) suppression of the structural order parameter of a two-dimensional charge-density wave (CDW) that clearly decouples from its electronic counterpart following fs optical quenching. Through atomic fluctuational analysis on Bragg reflections and satellite features, we identify important momentum-dependent electron-phonon couplings appearing on both time scales that may be related to interactions between the unidirectional CDW collective modes, the lattice phonons, and the perturbed electronic subsystem. We show that the characteristic time scales of these couplings and relative fluctuational amplitudes as characterized by fs crystallography jointly determine the cooperativity between the electronic and structural subsystems, and from this it is possible to elucidate the underlying mechanism of the charge-ordered system.Physical review. B, Condensed matter 08/2012; · 3.66 Impact Factor - SourceAvailable from: export.arxiv.org[Show abstract] [Hide abstract]

**ABSTRACT:**Rare-earth tri-tellurium RTe3 is a typical quasi-two-dimensional system which exhibits obvious charge-density-wave (CDW) orders. So far, RTe3 with heavier R ions (Dy, Ho, Er, and Tm) are believed to experience two CDW phase transitions, while the lighter ones only hold one. TbTe3 is claimed to belong to the latter. However, in this work we present evidences that TbTe3 also possesses more than one CDW order. Aside from the one at 336 K, which was extensively studied and reported to be driven by imperfect Fermi surface nesting with a wave vector q =(2/7c*), a new CDW energy gap (260 meV) develops at around 165 K, revealed by both infrared reflectivity spectroscopy and ultrafast pump-probe spectroscopy. More intriguingly, the origin of this energy gap is different from the second CDW order in the heavier R ions-based compounds RTe3 (R =Dy, Ho, Er, and Tm).Physical Review B 01/2014; 89(7). · 3.66 Impact Factor - L. Rettig, R. Cortés, J.-H. Chu, I. R. Fisher, F. Schmitt, P. S. Kirchmann, R. G. Moore, Z.-X. Shen, M. Wolf, U. Bovensiepen[Show abstract] [Hide abstract]

**ABSTRACT:**The femtosecond dynamics of the Fermi surface of DyTe3 and its band structure are investigated by time- and angle-resolved photoemission spectroscopy. We directly monitor the ultrafast collapse of the charge density wave gap within 200 fs.The European Physical Journal Conferences 03/2013; 41:03025-.

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ARPES Study of the Evolution of Band Structure and Charge Density Wave Properties

in RTe3 for R = Y, La, Ce, Sm, Gd, Tb and Dy

V. Brouet1,2,3, W.L. Yang2,3, X.J. Zhou2,3, Z. Hussain2, R.G. Moore3,4, R. He3,4, D.H. Lu3, Z.X. Shen2,3,4,

J. Laverock5, S. Dugdale5, N. Ru4 and I.R. Fisher4

1Lab. Physique des Solides, UMR8502, CNRS, Université Paris-Sud XI, Bât 510, 91405 Orsay (France)

2Advanced Light Source, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA

3Stanford Synchrotron Radiation Laboratory, Stanford University, Stanford, California 94305, USA

4Geballe Laboratory for Advanced Materials and Department of Applied Physics, Stanford University,

Stanford, California 94305-4045, USA

5H. H. Wills Physics Laboratory, University of Bristol, Tyndall Avenue, Bristol BS8 1TL, United Kingdom

We present a detailed ARPES investigation of the RTe3 family, which sets this system as an ideal

"textbook" example for the formation of a nesting driven Charge Density Wave (CDW). This family

indeed exhibits the full range of phenomena that can be associated to CDW instabilities, from the opening

of large gaps on the best nested parts of Fermi Surface (FS) (up to 0.4eV), to the existence of residual

metallic pockets. ARPES is the best suited technique to characterize these features, thanks to its unique

ability to resolve the electronic structure in k-space. An additional advantage of RTe3 is that the band

structure can be very accurately described by a simple 2D tight-binding (TB) model, which allows one to

understand and easily reproduce many characteristics of the CDW. In this paper, we first establish the main

features of the electronic structure, by comparing our ARPES measurements with Linear Muffin-Tin

Orbital band calculations. We use this to define the validity and limits of the TB model. We then present a

complete description of the CDW properties and, for the first time, of their strong evolution as a function

of R. Using simple models, we are able to reproduce perfectly the evolution of gaps in k-space, the

evolution of the CDW wave vector with R and the shape of the residual metallic pockets. Finally, we give

an estimation of the CDW interaction parameters and find that the change in the electronic density of states

n(Ef), due to lattice expansion when different R ions are inserted, has the correct order of magnitude to

explain the evolution of the CDW properties.

I. INTRODUCTION

Charge density waves (CDWs) are typical

instabilities of the Fermi Surface (FS) in the presence of

electron-phonon coupling.1 They occur when many

electrons can be excited with the same q vector of one

particular phonon mode at a moderate energy cost, i.e.

by keeping these electrons near the Fermi level. The

ideal case is when all electrons can be excited this way,

which implies that all parts of FS can be connected by q

to some other parts. This property of the FS is called

perfect nesting. For an ideal one dimensional (1D)

system, the FS consists of two points, at –kf and +kf, so

that it exhibits by definition perfect nesting at q=2kf.

Hence, all 1D systems are subject to CDW transitions,

often called Peierls transitions in this case. In real

systems, the good nesting properties are usually reduced

to some particular regions of the FS, but are nevertheless

often sufficient to trigger CDWs (or Spin Density Waves

(SDWs), their spin analog) in many quasi-1D, 2D or

even 3D systems (a famous example is the SDW of Cr2).

In this case, the CDW/SDW gap is expected to open

only on the best nested FS parts and the system may

remain metallic in its broken symmetry ground state.

Although these ideas were introduced in the 1950s

and CDW were heavily studied experimentally since the

mid-1970s,3 they remain a subject of interest for today's

research, because of the continued interest in low

dimensional systems. Such systems indeed offer

fascinating opportunities to study new states of matter,

where electronic correlations probably play a major role,

because the spatial confinement enhances the probability

of interactions. In these systems, CDWs or SDWs are

often instabilities competing with more exotic ground

states, and they are therefore important to fully

characterize. For example, Yao et al. used RTe3 to study

the competition between checkerboard and striped

charge orders, which may be of relevance for

comparison with cuprates or nickelates.4

Also, the technical progress of Angle Resolved

Photoemission (ARPES) in the past 2 decades, has made

it possible, in principle, to illustrate very elegantly the

impact of the CDW formation on the electronic

structure, in a much more direct way than with any other

experimental methods. As ARPES produces images of

the FS, one can directly examine its nesting properties

and compare them with the strength of the CDW gap

measured on the different FS parts. Despite this, there

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are not many examples of CDW systems, where ARPES

could be used to illustrate all these points. This is mainly

because finding a truly low dimensional system that

remains relatively simple is a rather difficult task. As we

have seen, quasi-1D systems are the most likely hosts for

CDW. However, they also exhibit by nature serious

deviations from the Fermi-liquid theory that complicates

ARPES analysis. In particular, Fermi edges are usually

not well defined, making the definition of gaps more

difficult, as in organic conductors,5 (TaSe4)2I6 or

K0.3MoO3.6,7 Also, for a truly 1D system the gap would

open homogeneously on the FS, which reduces the

interest of a k-resolved probe like ARPES. In the quasi-

1D system NbSe3, imperfect nesting gives rise to

coexistence between gapped and metallic regions, which

could be observed in ARPES8, despite a rather

complicated band structure. Hence, quasi-2D systems

appear as a more simple choice for photoemission

studies. Transition metal chalcogenides (1T-MX2 or 2H-

MX2 with M=Ti, Nb, Ta and X=S, Se, Te) exhibit a

variety of quasi-2D CDW behaviors that have been

extensively studied with ARPES.5,9 Despite this, no

simple relation between nesting properties and the

reported gaps (ususally 10-20meV) could be firmly

established9,10 and the mechanism of the CDW itself is

still debated.11 Let us emphasize that despite the

presence of chalcogenides, these systems have little in

common with RTe3. Triangular planes of transition metal

ions dominate their electronic properties, while these are

square planes of Te in RTe3. To conclude this short

overview of ARPES in CDW systems, let us mention the

case of the surface CDW in In/Cu(001),12 which exhibits

interesting similarities with RTe3, although it is not a

bulk phase transition.

RTe3 are quasi-2D metals, where a much clearer

situation is encountered. We will argue that they do

allow the illustration of the main CDW concepts fairly

well and further raise interesting questions on the limits

of different CDW models. After the seminal work

initiated by DiMasi et al.13-15, there has been a growing

wealth of information about these materials gathered

through detailed structural16,173, STM16,18, ARPES19,20,

transport21 and optical studies22,23. It was first believed

that these compounds always remain in the CDW state,

up to their melting point, making the language of phase

transition questionable. We have recently revealed a

transition to the normal state at 244K in TmTe3 up to

416K in SmTe3 and presumably even higher temperature

for lighter R.17 This definitely qualifies this family as

CDW materials and opens new perspectives for a full

characterization of the CDW state, including the

fluctuations above the transition. In this paper, we will

restrict our study to the characterization of the ground

state of the light rare-earths, from LaTe3 to DyTe3. For

the heavy rare-earth (DyTe3 and above), two successive

phase transitions occur,17 which we do not consider here.

The possibility of tuning the CDW properties (the

transition temperature, the size of the gap, the wave

vector qcdw) with R is a rather unique property of this

family, which is very useful to discuss the origin of the

CDW. A similar variation of the CDW properties can be

induced by applied pressure,23 making it likely that the

changes are due to lattice contraction.

The CDW in RTe3 is characterized by large

displacements (about 0.2Å)16 and large gaps in the

electronic structure (up to 0.4eV).19,20,22 This large gap is

an advantage for ARPES studies, because it makes it

easy to measure its location and changes in k-space

accurately. On the other hand, it raises questions about

the nature of the CDW and especially whether the

traditional weak coupling treatment of the nesting driven

CDW would still hold. Indeed, the gap is presumably

several times larger than the phonon frequencies

involved. In such a situation, a strong coupling model of

the CDW could appear more appropriate, where the

structural distortion is really the driving force for the

transition. The local tendency of Te atoms to form stable

chemical bonds would be its starting point. Indeed, a

usual Te-Te bond is 2.8Å, whereas the average distance

between Te in the planes hosting the CDW is about

3.1Å. Whangbo and Canadell discussed, in the case of

1T- or 2H-MX2, similarities and differences between the

approach of FS nesting or chemical bonding.24 The

distinction between weak and strong CDWs was

investigated by Nakagawa et al. in their study of

In/Cu(001).12 They concluded the CDW in this system

was of "dual nature". In this paper, we will show, with

unprecedented details for any CDW system, that the

predictions of the FS nesting scenario explain extremely

well the openings of gaps observed by ARPES in RTe3.

Furthermore, the main variations of the CDW properties

with R can be well explained by an additional

stabilization of the CDW due to the enhancement of

n(Ef) through lattice contraction. This gives substantial

ground that the electronic energy is, at least, an essential

ingredient for the formation of the CDW in RTe3.

Large single crystals of RTe3 were grown by slow

cooling of a binary melt.21 The crystals easily cleave

between two Te planes, providing a good surface quality

for ARPES. ARPES measurements were mostly carried

out at the beamline (BL) 10.0.1 of the Advanced Light

Source (ALS), with a Scienta-2002 analyser, an energy

resolution better than 20meV and an angular resolution

of 0.3°. Other data were acquired at BL 12 of the ALS

(Fig. 7, Fig. 12) and BL 5-4 of the Stanford Synchrotron

Radiation Laboratory (Fig. 11). All measurements were

performed at low temperatures T≈20K.

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II. TIGHT-BINDING MODEL OF THE

ELECTRONIC STRUCTURE

In RTe3, Te planes are stacked with R/Te slabs,25 as

sketched in Fig. 1a. Note that we follow the usual

convention where the b axis is perpendicular to the Te

planes (b≈26 Å). The planar unit-cell (a,c) is defined by

the R atoms of the slab (orange square in Fig. 1a and 1b).

There is a small orthorhombic distortion of this square,

which shrinks from a=4.405Å and c=4.42Å in LaTe3 to

a=4.302Å and c=4.304Å in DyTe3.17 The Te atoms in the

planes form a nearly square net, but with a square unit

(green square) rotated by 45° with respect to the unit cell

and with only half the area. Hence, two different

Brillouin Zones (BZ) will be convenient to use

throughout this paper : a 2D BZ built on the Te square

from the plane and the 3D BZ built on the lattice unit

cell (see Fig. 1c). We define a*=2π/a and c*=2π/c as unit

wave vectors of the 3D BZ.

kkE

x

−=

2),(

This model is plotted on top of the calculated band

structure in Fig. 2a, as red and blue lines for px and pz,

respectively. As the TB bands are constructed for one Te

plane, they have the periodicity of the 2D BZ and they

have to be folded back with respect to the 3D BZ

boundaries to acquire the 3D lattice symmetry. These

The band structure was calculated using the linear

muffin-tin orbital (LMTO) method for a fictitious LuTe3

composition and a=c=4.34Å.26 Lu was chosen to avoid

the complications associated with the description of f

electrons in the local-density approximation. The

calculated band structure is shown in Fig. 2a along c* at

a fixed kx=0.3a* and ky=0. Eighteen different bands are

found between 2eV and –6eV, corresponding to the Te

5p orbitals of the 6 Te per unit cell (2 in the slab and 4 in

the planes). However, only 4 bands cross the Fermi

level, corresponding to the Te in-plane px and pz orbitals.

They are well isolated from other bands over a 1eV

window below Ef.

The dispersion of these bands can be very well

reproduced by a tight-binding (TB) model of the Te

plane. We consider only the two perpendicular chains of

px and pz orbitals, represented on Fig. 1b in red and blue,

with a coupling tpara along the chain and tperp

perpendicular to the chains. We assume a square net and

totally neglect the coupling between px and pz. Using the

axes of the 3D BZ, this yields the following dispersions.

[

[

]

]

[

[

]

]

fzx perpzx parazxp

fzx perpzxparazxp

EakktakktkkE

Eakktakkt

z

−+−−

−

−−+−=

2/*)(cos*22/*)(cos*

2/*)(cos*22/*)(cos*2),(

additional folded bands are shown as dotted lines in Fig.

2a. The Fermi level Ef =-2tparasin(π/8) was fixed so that

Fig. 1 : (a) Sketch of the RTe3 structure. (b) Sketch of the Te plane (green

points) with in-plane px and pz orbitals in red and blue. The 3D (a,c) unit

cell is shown as orange square. (c). Sketch of the reciprocal space with

the 3D BZ (orange) and the 2D BZ (green) that would correspond to one

isolated Te plane.

Fig. 2 : (a) Band Structure along c* for kx=0.3a* and kz=0

calculated with LMTO method. Red and blue lines are TB fits for px

and pz. Dotted lines are folded bands. (b) Zoom-in of the electronic

structure near Ef measured with ARPES in CeTe3 at 55eV along the

same direction. Red lines are the calculated bands.

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px and pz each contain 1.25 electrons. We assume here

that all R are trivalent15,21 and donate 2 electrons to each

Te in the slab and 0.5 electrons to each Te of the 2

planes. We further assume that the out-of-plane Te

orbital py is completely filled, leaving 2.5 electrons for px

and pz. The TB parameters were adjusted to reproduce

best the calculated band structure, which is reached for

tpara=-1.9eV and tperp=0.35eV. It is worth noting that

although |tperp|<<|tpara|, it is much larger than the

temperature (tperp≈3000K), so that one would not be in a

1D limit, even for an isolated set of chains. As there are

two Te planes per unit-cell, the bands in the calculation

are doubled and there is a clear "bilayer splitting"

between them. In this calculation, spin-orbit couplings

were neglected.

Fig. 2b shows the corresponding electronic structure

measured with ARPES. The agreement with the

calculation for the bands at the Fermi level is very good,

except the intensity of the folded bands is so weak that

they are hardly distinguishable. As explained in ref. 20,

the intensity of folded bands is very generally

proportional to the strength of the coupling responsible

for the folding. The weak intensity of the folded bands

reflects here the small 3D couplings and consequently,

the nearly 2D character of these compounds. For the

deeper Te bands, we observe some deviations between

the calculated and measured bands, more details will be

given in part II.C.2.

The excellent description of the electronic structure

near Ef with only in-plane Te orbitals suggest a

negligible coupling with the R/Te slab. The transport

anisotropy is indeed very large, at least a factor 100.15,21

In this case, one expects that the main consequence of

changing R will be a change in bandwidth due to the

expansion or contraction of the Te square lattice. Fig. 3

displays the ARPES intensity of the Te bands integrated

around the Γ point, for different rare-earths. Their

structure is quite similar, confirming the small influence

of rare-earth orbitals. The total bandwidth can be

estimated by the peak position of the last band. It

increases from about 4.25eV for Ce (c=4.385Å) to

4.70eV in Gd (c=4.33Å) and Tb (c=4.314Å). This is in

qualitative agreement with the larger overlaps between

Te orbitals expected for smaller lattice parameters.

In our calculation, the bandwidth at Γ increases

when expanding the lattice from 4.75eV (La, c=4.42Å)

to 5.15eV (Dy, c=4.03Å), i.e. by 8%. While these

absolute values are a little larger than the experimental

ones, the order of magnitude of the expansion is in good

agreement. This corresponds in the calculation to a

decrease of the density of states at the Fermi level from

n(Ef)=1.6states/eV/cell (La) to 1.5states/eV/cell (Dy).

With the TB model, for tpara=-1.9eV and tperp=0.35eV

estimated before for a=c=4.34Å,

n(Ef)=1.48states/eV/cell, slightly smaller. This is the

same trend as for the bandwidth and suggests that a

slightly smaller value of tpara might be more appropriate

to describe RTe3. We observe that n(Ef) solely depends

on tpara for realistic values of tperp. To reproduce the

calculated n(Ef) values, one has to use tpara=-1.7eV (La)

to –1.85eV (Dy). This is a variation of 8%, in good

agreement with that of the bandwidth, both calculated

and experimentally observed. Therefore, we will use this

parameter range in the rest of the paper to model the

changes of the electronic structure from La to Dy.

A. Fermi Surface

The FS expected in the TB model is very simple. It

is made out of two perpendicular sets of nearly parallel

lines, corresponding to the two chains. They are shown

in Fig. 4a in red and blue, for px and pz respectively.

With no perpendicular coupling (tperp = 0), the problem

would be reduced to that of two perfectly 1D chains,

perpendicular to each other, and the FS would consist of

two sets of exactly straight lines, exhibiting perfect

nesting.14 The coupling between the chains introduces a

deviation from one dimensionality and a curvature of the

FS proportional to |tperp/tpara|. We will show in part III

(e.g. Fig. 17) that it is precisely this curvature that makes

the nesting imperfect. The orange arrow indicates the

best nesting wave vector qN=0.68c*. There are other

wave vectors giving better (actually perfect) nesting for

px or pz, but this one reaches a better compromise by

nesting equivalently px and pz. The competition between

these different wave vectors has been studied by Yao et

al.4

we calculate

Fig. 3 : Te valence bands integrated around the Γ point, measured

for different rare-earths, at a photon energy of 55 eV.

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The TB FS in Fig. 4a is shown for the two sets of

parameters corresponding to Dy and La. The two

contours are however so close that they cannot be

distinguished. This is normal, as the FS area should be

proportional to the number of holes in the band, which is

kept constant in our model. When the band width

changes, the Fermi level readjusts to keep this area

constant. Therefore, the FS contour is independent of

tpara, as long as its shape (i.e. |tperp/tpara|) remains the same.

Interestingly, this means that, in this model, the nesting

properties remain exactly the same throughout the series.

The experimental FS of RTe3 was first measured by

Gweon et al. in SmTe3.19 In Fig. 5, we present the FS for

different R = Y, Ce and Sm, compared with the

predictions of the TB model. FS are obtained by

integration of the spectral weight in a 10meV window

around Ef. The spectral weight is strongly suppressed in

a large region around kx=0 (note that the axes are rotated

for YTe3). We will show in part III that this is due to the

opening of a large CDW gap in these regions. Although

the directions of kx and kz appear at first quite similar in

structure, x-ray measurements have shown that the gap

always open along the c axis.17 Accordingly, we only

observe the gap opening along kz. These regions are the

best nested ones (see Fig. 17), in those with poorer

nesting, the gap does not open at the Fermi level and we

observe again intensity at Ef. Interestingly, the ungapped

regions appear larger in YTe3 and SmTe3 than CeTe3

(this is particularly clear for the ungapped fraction of the

square). We will analyse this behavior quantitatively in

part III and show that it can be understood from the

larger gap of CeTe3.

Clearly, the distribution of the spectral intensity is

equally well described by the TB model in the three

cases and this was true for all the rare-earths we have

measured. As discussed before, this does not give

information on tpara, but rather proves that there are no

significant changes in the band filling. This is not a

trivial result as it is for example not the case in RTe2.27

On the other hand, the well-defined curvature of the FS

allows us to estimate tperp=0.35± 0.08eV. This is a

totally independent estimation from the previous section,

but turns out to be in very good agreement.

The major deviation between the experimental FS

Fig. 4 : (a) Tight Binding FS for (tpara=-1.7eV, tperp=0.35eV) and (tpara=-1.85eV, tperp=0.35eV). The two contours overlap almost perfectly (see

text). The green square delimits the BZ corresponding to one Te plane. The orange arrow represents the best nesting wave vector qN. (b) Same

as (a) after interaction between px and pz that separates the "square" part of FS from the "outer" part. (c) Same as (b) plus, as dotted lines, the FS

contours folded with respect to the 3D BZ limits (green square). The orange arrow represents the equivalent nesting direction, but defined in the

3D BZ : (c*-qN).

Fig. 5 : Fermi Surface for YTe3, CeTe3 and SmTe3, obtained by integration of spectral weight in a 10meV window around Ef. No symmetry

operation were applied to the data. Photon energy was 35eV for YTe3, 55eV for CeTe3 and SmTe3, polarization was almost perpendicular to the

sample surface. Red and blue lines correspond to the Fermi Surface calculated with the TB model described in the text. Black contours on YTe3

map are guides for the eyes for the shape of square and outer pieces of FS. The suppression of spectral weight around kx=0 is due to the

opening of the CDW gap (note that YTe3 map is rotated by 90°).

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and the TB fit takes place at the crossing between px and

pz. This is because there is no coupling between these

bands considered in our TB model. Fig. 5 shows that, in

reality, they do interact and this rounds the FS contours

near the crossings. This effect is simulated in Fig. 4b : it

creates two different sheets of FS, a small hole-like piece

around Γ, called hereafter "square", and a larger

electron-like piece mainly in the second BZs, called

hereafter "outer". Note that the nesting quality and the

wave vector do not change when this interaction is

added, because the effect is symmetric on the square and

outer FS that are nested into one another.

In the 3D BZ, the folding of the FS gives rise to the

dotted black contours of Fig. 4c. Experimentally, Fig. 5

shows that their intensity is always very weak, except

near the zone boundaries.20 The nesting properties are

the same, except they would be described by (c*-qN) in

the 3D BZ.

B. Dispersion

In Fig. 2, one can check that not only the position of

the FS crossings but also the slopes of the dispersions are

in very good agreement with the TB model. This value

directly depends on tpara, so that its evolution as a

function of rare-earth may give additional insights into

the evolution of the electronic structure.

Metallic properties are best measured on the outer

part of FS, where the lineshape is more simple (see parts

C and D). Near Ef, the dispersion is nearly linear and we

extract its slope by a linear fit over a 0.2eV window. We

observe that the changes in the slope of the dispersion

are small along the outer FS and also as a function of R.

In fact, the slope essentially depends on the direction in

which the dispersion is measured, i.e. the angle α of the

detector slits of the analyser with respect to kz axis.

The TB model again offers a useful guide to

understand this evolution. As the bands are essentially

one dimensional, the slope of the dispersion is nearly

constant when measured along the chain direction (i.e.

α=± 45°). We define this value as the reference Fermi

velocity, which is

2

atV

hand, the slope of the dispersion rapidly falls to zero if

measured perpendicularly to the chain. This dependence

is illustrated on Fig. 6a for the two values of kx between

which the dispersion can be measured reliably on the

outer FS. It is nearly a cosine function of α and only the

small tperp gives it some kx dependence. Many different

values measured for different samples and/or branches of

the FS are reported as color points. The variation with kx

is indeed within the error bar of the measurement and the

general trend is the dependence with α. In Fig. 6b,

we plot Vf as a function of the lattice parameter c, after

correcting for the α dependence. We obtain an average

value for all samples Vf=10± 1 eV.Å. Within the TB

model, this corresponds to tpara=-1.7±0.15eV, a value that

corroborates our previous estimation. The variation

expected as a function of lattice parameter in the

previous paragraph (tpara=-1.7 to 1.85eV) is shown in

Fig. 6b to be within the error bar of the measurement.

Let us note that the dispersion is defined here over a

rather large energy scale and this analysis does not

exclude possible renormalization effects near the Fermi

level.

C. Deviations from the tight-binding model

1. Ce contribution

The smooth changes of the electronic structure as a

function of rare-earth is a good indication that they do

not play an active role in the electronic properties.

However, it would be interesting to clarify the

relationship between the localized 4f moments on the

rare-earth and the Te band, even if their coupling is

) sin(

ak

f paraf=

. On the other

Fig. 7 : Comparison of the electronic structure, measured at T=20K

and kx=0.6a*, in CeTe3at a photon energy of (a) 110eV and (b)

125eV, i.e. respectively off and on the Ce 4d-4f resonance. The two

parabola come from the main and folded outer Te bands (see Fig.8),

the line at –0.28eV in (b) is attributed to the spin-orbit satellite of

the Ce resonance.

Fig. 6 : (a) Color points : measured values of the slope of dispersion

for different RTe3 compounds and different branches of FS, as a

function of the angle α of the measurement. Black lines : variation

of the slope of the dispersion near Ef, calculated in the TB model,

for kx=0.6a* and 0.7a*, tpara=-1.7eV and tperp=0.35eV. (b) Vfvalues

corrected for the α dependence as a function of lattice parameter.

Black lines are theoretical variations of Vf for tpara=-1.7eV and –

1.85eV.

Page 7

7

weak. We present here results for CeTe3, for which the

magnetic susceptibility indicates localized moments of

2.4μB, consistent with trivalent Ce that order

antiferromagnetically at TN=2.8K.21,28 There is a mild

upturn of the resistivity below 10K, suggestive of a weak

Kondo behavior.21

In CeTe3, we observe a non-dispersive line at E=-

0.28eV throughout the whole BZ. In Fig. 7, we show that

it is strongly enhanced at the Ce 4d-4f transition

(120eV), indicating it has Ce character. This line is

indeed absent in other RTe3 systems. It does not interact

strongly with Te bands, as there is no detectable

perturbation of the Te dispersion at their crossings.

Generally, one would expect a two peak structure

for the Ce spectrum, corresponding to screened and

unscreened final states of the hole created through the

photoemission process (respectively 4f1 and 4f0).29 Their

positions and relative intensities are very sensitive to the

nature of the coupling between localized moments and

the metallic band. Such two peaks were observed in

CeTe2 at –4eV (4f0) and –1eV (4f1),30 which is typical of

a localized Ce3+ in a nearly insulating medium. In CeTe3,

the 4f1 peak moves closer to the Fermi level (-0.28eV)

and we did not resolve the 4f0 peak from other Te bands

at lower binding energies. The 4f1peak is known to

exhibit a spin-orbit splitting of 0.28eV between 4f1

4f1

observe is in fact the 4f1

peak centred at the Fermi level, but having a negligibly

small intensity.31 This is the situation expected at

temperatures higher than the Kondo temperature TK.29

As this measurement was done at T=20K, this result

corroborates the idea that CeTe3 is a weak Kondo

system, with TK<<20K.

2. Bilayer splitting

One thing neglected in the TB model is the coupling

between the Te planes that gives rise to the bilayer

splitting (see Fig. 2). Fig. 8 shows the FS of LuTe3

obtained with the LMTO method. The shaded area

corresponds to the gapped area. The amplitude of the

bilayer splitting is indicated as color scale. It changes

quite strongly along the FS, it is larger in the square

(δ=0.03c∗) than in the outer part (typically less than

δ=0.01c*). The typical full width of our spectra at half

maximum are found between 0.02 and 0.03c*.

Consequently, bilayer splitting is usually not resolved on

the outer Fermi Surface (except near the corners) but it is

in the square. In Fig. 9 and 10, we give examples of the

typical lineshapes observed in these two parts.

In the square, we typically observe two lines

7/2 and

5/2 states. This suggests that the satellite line we

5/2 spin-orbit satellite of a 4f1

7/2

reaching for Ef. Fig. 9 gives two examples at different

heights in the square, one metallic (Fig. 9a) and one

gapped (Fig. 9b). They form a sort of inverted V-like

shape that is tempting to attribute to the bilayer splitting,

although, in the calculations, the two lines are more

parallel, at least near Ef. The relative intensities of the

two lines are very sensitive to the photon energy, they

are quite different at 55eV (Fig. 9a) but nearly equal at

35eV (Fig. 9b). Such oscillations in the intensity of

bilayer split bands with photon energy were also

observed in the well studied case of Bi2Sr2CaCu2O8+δ.32

The curvature of the outside band seems to be due to the

crossing with the folded band at –0.8eV (see also Fig. 2),

but it appears more pronounced experimentally

(especially in Fig. 9a).

Other bands are present within the square

corresponding to different Te orbitals. The agreement

with the calculation is not as good as for the near Ef

Fig. 9 : Near Ef band structure in the square. (a) Measured in

CeTe3 with photon energy of 55eV at kx=0.23a* and (b) in TbTe3

with 35eV and kx=0.1a*. Red lines are calculated dispersions.

Fig. 8 : LuTe3 FS calculated with the LMTO method. The color

scale indicates the amplitude of bilayer splitting. The shaded area

corresponds to the part of FS gapped by CDW. The red lines

correspond to cuts used to give example of bilayer splitting, in Fig.

9 and 10. Other lines correspond to cuts shown throughout this

paper in figures with number as indicated.

Page 8

8

bands, which is probably a consequence of the

approximations used in the LDA. We frequently observe

shifts of deep Te bands to lower binding energies. This is

the case in Fig. 9b, where a band with a distinctive two

lobe structure seems pushed along the dotted arrow by

about 0.5eV. These shifts seem common, as a similar

trend was reported in LaTe2.33

Fig. 10 shows a typical situation for the outer FS.

Near the corner (kz=0), the bilayer splitting is maximum.

In Fig. 10a, the two lines are well resolved, they have

almost the same intensity because the photon energy is

35eV, and the bilayer splitting is 0.025a* near Ef. At

higher kz, the bilayer splitting rapidly decreases and is

not resolved anymore along most of the outer FS, as

shown in Fig. 10b for kz=0.32c* (see position in Fig. 8).

The linewidth of the main line is here δν=0.025a*,

indeed quite larger than the calculated splitting at this

position, δ=0.005±0.002a* (the error bar includes

differences depending on the details of the calculation).

It is also unlikely that this width is dominated by the

bilayer splitting, as the lines are not narrower when it is

resolved, as in the square (δν=0.024c* in Fig.9a) or the

outer corner (δν=0.035a* in Fig. 10a).

3. 2D character

An interesting issue is the strength of the 3D

couplings in these quasi-2D systems, which we have

implicitly neglected so far. Although the transport

anisotropy is large, these systems remain metallic along

the b-axis15,21 implying sizable hybridization in the

perpendicular direction. 3D couplings often complicate

the analysis of the photoemission spectra and might be

responsible for a residual broadening of the spectra.34

Their order of magnitude is therefore important to

evaluate.

Fig. 11a shows the dispersion of Fig. 2 calculated

for different values of ky. The value of k⊥ in an ARPES

measurement is not known precisely, because it is not

conserved at the surface crossing. It depends on the

photon energy Eph ; at the Γ point, k⊥=

where V0 is an inner potential adjusted experimentally.35

One can therefore expect a line to shift and/or broaden as

a function of the photon energy on the energy scale of

the perpendicular dispersion. The relative amount of

broadening and shift will depend on the lifetime of the

photoelectron in the final state.34 Indeed, the linewidth

can be written in simple cases as δν=Γi+vi⊥/vf⊥ Γf, where

Γi and Γf are the lifetimes in the initial and final states,

vi⊥ and vf⊥ are the slopes of the perpendicular dispersion

in each state.36 One is typically interested in Γi, but it is

usually masked by the larger Γf (the final state having a

much higher energy, of the order of the photon energy),

unless vi⊥ is very small.

For the outer bands, the dispersion is fairly

independent of ky, as expected for a good 2D metal. The

total spread of kf as a function of ky is δ3D=9.10-4c* and

δ3D=2.10-4c*, for the two split bands, obviously totally

negligible compared to the width of δν=0.02-0.03c*.

The perpendicular dispersion is larger in the square, as

could already be anticipated from the larger bilayer

splitting, indicative of stronger transverse couplings. Fig.

11b displays the variations of kf expected theoretically,

for the red and blue bands forming the square (solid

lines). They are plotted as a function of photon energy,

assuming a typical value V0=10eV 34,35 in the previous

formula of k⊥. They are compared with kf values

measured in CeTe3 at different photon energies (black

points). Note that the bilayer splitting is not resolved in

the square near Ef (Fig. 9a). We do not observe

oscillations in the position measured for kf and the

variation is less 0.007c*. Although the linewidth is of the

same order of magnitude as the calculated perpendicular

dispersion for the red band, it is unlikely that it is

dominated by Γf, since the linewidth is very similar for

the outer band, where vi⊥ is reduced by a factor 20. We

conclude that there are no obvious contribution of 3D

couplings detectable in our spectra.

0

2

2

VEm

ph+

h

,

Fig. 10 : Near Ef band structure along the outer FS measured in

YTe3 in the 2nd BZ (see Fig. 8) at 35eV, for (a) kz=0.05c* and (b)

kz=0.32c*.

Fig. 11 : (a) Calculated band dispersion at kx=0.3a*, for different ky

values. (b) Black points : kf position measured in CeTe3 for the

square as a function of photon energy. Solid lines : perpendicular

dispersion expected theoretically on the square for the red and blue

bands of pannel (a).

Page 9

9

D. Spectral lineshape

Because of its simple, well understood and nearly

2D electronic structure, RTe3 offers a favorable situation

to extract detailed information about the electronic self-

energy from the ARPES lineshape.32,35 Since examples

of "simple" low dimensional systems are rare, this

deserves attention. Fig. 12 exemplifies typical lineshapes

along the outer FS, where the band is well separated

from other bands, the bilayer splitting is minimum and

3D couplings are negligible. They have low backgrounds

both for cuts taken at constant energy (MDC) and

momentum (EDC), which is another advantage for line

fitting. Yet, the linewidth (Fig. 12c) appears rather large

and does not exhibit a significant decrease near Ef, which

would be the fingerprint of a Fermi liquid.37,38 We note

however that the Fermi step is very well defined (see

also Fig. 21), so that this case is completely different

from that of a "bad metal", where broad lines are

associated with a low weight at the Fermi level, due to

strong correlations, low dimensionality and/or polaronic

effects.5 It is worth emphasizing this point, because the

opening of the CDW gap along c* gives an effective 1D

character to the problem, which raises the question of

possible 1D features in the physics of RTe3. A power

law behavior of the optical conductivity was recently

attributed to the formation of a Tomonaga-Luttinger

liquid.22 We do not observe equivalent effects in the

ARPES lineshapes.

The behavior of the width in Fig. 12 is typical of

that found in all RTe3 systems we have investigated.

Most of the changes we observed as a function of

binding energy could ultimately be attributed to

crossings with other lines (such as the Ce satellite at –

0.28eV) or weak CDW shadow bands (red arrow in Fig.

10b, see III.B). The MDC linewidths are comprised

between δνa=0.03 and 0.05 Å-1

and/or different cleaves. This translates to a rather large

energy width for EDC spectra (δνe= δνa*Vs = 0.2-0.3eV,

where Vs is the slope of the dispersion), but this is

essentially due to the fact that Vs is about one order of

magnitude larger here than in many reference systems,

such as cuprates32 or 1T-TiTe2.38

If taken at face value, the MDC width would

correspond to a mean free path l=2/δνa=40-60 Å, which

seems rather small for these good metals where quantum

oscillations have been observed.39 As transport and

ARPES lifetimes are different, this comparison is only

qualitative. We have seen that the bilayer splitting and

3D couplings should be rather negligible contributions

on this scale. On the other hand, the angular resolution

was usually set to δkres=0.3°, which is smaller but not

for different samples

negligible compared to these linewidths (it corresponds

to 0.02Å-1 at 50eV). For these systems with large Fermi

velocities, the angular resolution is indeed a much

stronger constraint than energy resolution (δEres was

typically 10-20meV, much smaller than Vs * δkres). We

did not observe a large improvement when using a

higher resolution mode of the analyser (0.1°), which

rules out that it is entirely a resolution problem. We

believe that other types of extrinsic angular broadening

could be a problem, for example a distribution of angles

at the sample surface. Although these samples cleave

very well, giving smooth and shiny surface, they often

exhibit curved surfaces, which could limit our effective

resolution. ARPES experiments with ultra small spot

may be able to clarify this issue. Alternatively,

impurities at the surface could reduce the mean free path

we measure.

III. CDW PROPERTIES

Fig. 13 and 14 summarize the main results of our

ARPES investigation of the RTe3 CDW properties. The

evolution of the gap in k-space is shown in Fig. 13 for

different compounds. We find identical gaps on the

square and outer FS pieces for a given kx value, therefore

the gap is plotted as a function of kx. The gap is

maximum at kx=0 and decreases to zero for a value kx

comprised between 0.18 and 0.28 a*. This qualitative

behavior is the same for all rare-earth we have studied

(Dy, Tb, Gd, Sm, Ce, La) and Y. On the other hand,

there are significant quantitative changes in the

maximum gap value and in kx

displays the maximum gap value as a function of the

lattice parameter. It is defined with respect to the leading

0

0

as a function of R. Fig. 14

Fig. 12 : Lineshape analysis in CeTe3 at 55eV and 20K for a cut of

the outer FS at kx=0.62c*. (a) EDC stacks. (b) Image plot. (c) Full

width at half maximum of MDC spectra (the value in energy is

obtained by multiplying with the slope of dispersion).

Page 10

10

edge of the spectra (see part III.D). In SmTe3, our value

(280meV) corresponds well to the 260meV first

measured by Gweon et al..19 The increase is roughly

linear as a function of lattice constant, with maybe an

exception for YTe3, which is the only non rare-earth

compound.

The change of the gap along the FS is an essential

feature to understand the origin of the CDW. We will

show that the model of a nesting driven sinusoidal

charge density wave allows to explain in details the

location of the gap in k-space. This model also implies

the presence of metallic pockets and shadow bands in the

regions that are not perfectly nested, which we observe

and will discuss. The increase of the gap with lattice

parameter is another way to investigate the nature of the

CDW. The basic idea is that the larger n(Ef) caused by

the lattice expansion should stabilize the CDW.13 We

give a quantitative discussion of this phenomenon, which

leads to estimation of the important CDW parameters,

such as the electron-phonon coupling and the relevant

phonon frequencies.

A. Interacting band structure in the CDW state

We introduce here a simple theoretical model to

describe the modification of the band structure in the

CDW state. The electron-phonon coupling responsible

for the CDW is described by the following hamiltonian.

∑∑

qkk

,

where gq is the electron-phonon coupling strength for the

wave vector q, ck

operators for electrons and aq

+

−

+

k

+

+

k

++=

qqkqqkk

aaccgccH

)(

ε

+

(ck) are creation (destruction)

+

(aq) for phonons.

In the CDW state, a static distortion takes place for a

wave vector ± qcdw , implying <a±q> = <a+

wave vector. This creates a coupling

±q> ≠ 0 at this

g

2

=

cdwcdw

qq

aV

between states k and

cdw

q

±

k

. New wave functions

allowing for the admixture of these states have to be

defined with the form :

uqku

cdw

+−=

−

ψ

where the coefficients are solutions of the matrix

⎤

⎢

k

V

ε

0

cdwqkk cdwqkk

qkuk

cdw

++

+

⎥

⎦

⎥

⎥

⎢

⎣

Here, we truncate the interaction at the first harmonic,

although, in principle, all harmonics n.qcdw should be

included.

The dispersion of these new wave functions are

shown in Fig. 15a, with a size of the markers chosen as

2

k u

. Wave functions are essentially unchanged away

⎢

⎡

+

−

V

cdw

q

cdw

q

k

k

V

V

ε

ε

0

.

Fig. 15 : (a) Red points : sketch of the dispersion in the CDW state at

kx=0, calculated for V=0.4eV and qcdw=0.32c*. The size of the points

is proportional to the spectral weight |uk|

line is the original dispersion from the TB model, dotted lines are

translated by c*±qcdw. (b) Zoom near the Fermi level for the same

calculation and different kx values.

2 (see text). The solid black

Fig. 13 : k-dependence of the gap along the Fermi Surface for the

square part of FS (red points) and outer part (blue points). kxis used

as implicit parameter for the position on FS. The black line describes

the decrease of the gap expected because of the imperfect nesting

away from kx=0 (see part III.D).

Fig. 14 : Maximum gap value for different rare-earths and YTe3

(blue point), plotted as a function of the lattice parameter c. The

straight line is a guide for the eyes.

Page 11

11

from the crossings between

k (solid line) and

cdw

q

±

k

(dotted lines), i.e. uk =1 or 0. At the crossing,

a gap of amplitude 2V opens and some weight is

distributed on the translated parts of the band structure

(dotted lines), which are called shadow bands. Their

intensity is proportional to V and decreases very fast

away from the crossing. This is better seen in Fig.15b,

where the part near the Fermi level is emphasized.

An example of the typical resulting shape of the

dispersion in the gapped state is shown in Fig.16. The

band "turns away" from the Fermi level after reaching a

maximum, corresponding to the gap value (here

330meV) and its intensity rapidly vanishes. The

observation of such a shape is in fact the best proof that

the band is indeed gapped versus its intensity would be

accidentally reduced near Ef by matrix element effects.

In Fig. 16b, we display its intensity as a function of the

distance with respect to the crossing between the main

and translated bands. The qualitative variation of this

intensity is in good agreement with the expectations of

the theoretical model. Two theoretical variations are

given for V=0.4eV (solid line) and V=0.2eV (dotted

line). However, a quantitative comparison remains

difficult, because, in many cases, the photoemission

intensity is modulated by matrix element effects that

partially mask the intrinsic variation.

Fig. 15b summarizes the evolution expected as a

function of the degree of nesting of the FS. When the FS

is perfectly nested, the crossing takes place by definition

at the Fermi level. The gap measured by ARPES with

respect to Ef is maximum and equal to V. This is the case

for kx=0 represented in red on Fig. 15b. When the perfect

nesting is lost (kx>0), the crossing takes place above Ef

and the apparent gap at Ef decreases (see kx=0.2).

Eventually, when the crossing takes place at an energy

higher than V above Ef, the band crosses again the Fermi

level and remains metallic (see kx=0.5). In this case, two

crossings should in fact be observed at Ef, with large and

small weights, forming the two sides of a metallic

pocket. These theoretical pockets are shown in Fig. 17

for the simplified case of an isolated px band. The side

with large weight follows the original FS and the other

side the FS translated by c*±qcdw. Note that we use here

the usual definition of qcdw in the 3D BZ, which in fact

nests the main and folded FS (see Fig. 4). We however

show translated bands by c*±qcdw, which is equivalent

but connects bands from the same Te plane. The

extension of the metallic pocket sensitively depends on

the strength of the gap, it is shown for V=0.2eV in Fig.

17a and V=0.4eV in Fig. 17b.

B. Value of the interaction parameter V

In the case represented in Fig. 17, the CDW vector

has been chosen, so that the original FS and its translated

exactly overlap on the corner of the square. In the

corresponding band structure (Fig. 15), the Fermi level at

kx=0 lies in the middle of the full interacting gap 2V.

However, as ARPES only measures occupied states, one

cannot directly observe this full interacting gap and this

leaves an ambiguity on the size of V. When kx increases,

the symmetry is such that the crossing between the main

and translated bands always takes place above the Fermi

level (Fig. 15b), so that the full interacting gap is still

eluding measurement.

Fig.16 : Shadow band in the gapped state. (a) Band from the

outer FS, measured along kz at kx=0.1a* in CeTe3. (b) Red points

: intensity of this band as a function of k (zero is the position of

the crossing). It is extracted by a fit of the EDC spectra with a

gaussian and a parabolic background. Theoretical variation for

this weight is also shown for V=0.4eV (as solid line) and

V=0.2eV (as dotted line).

Fig. 17 : Metallic pockets from the px band expected after CDW

reconstruction, for two different values of V. The interaction with

pzis neglected. Thick red and blue lines correspond to original

FS for px and pz. Thin lines are FS translated by c*+qcdw or c*–

qcdw. The size of the markers is proportional to the spectral

weight.

Page 12

12

A closer inspection of Fig. 17 reveals another type

of crossing between main and translated bands, indicated

by a circle. These bands belong to different orbitals, they

do not nest the original FS, but they link k and

+

k

electrons, and are therefore also subject to the

interaction V. Their crossing turns out to take place

below Ef, which allows to directly measure 2V. Indeed,

we see these bands interact with each other in our data,

as shown in Fig. 18. In Fig. 18a, a strong "break" is

observed in the dispersion of LaTe3, at a position

corresponding well to that expected for such a crossing.

The shadow band intensity is however so weak that one

could not guess there is a crossing there without the TB

simulation. The lineshape is detailed in Fig. 18b,

showing the opening of a break with peak-to-peak

distance 2V=0.37eV. The same behavior is observed for

YTe3 in Fig. 18c (same data as Fig. 10b). Interestingly,

the break is smaller in this case (2V=0.21eV). This is

consistent with the smaller gap in YTe3 (Δmax=0.33eV)

than in LaTe3 (Δmax=0.4eV). On the other hand, we

observe here Δmax≈2V rather than Δmax=V expected in

Fig. 15.

One would obtain Δmax=2V, if the Fermi level at

kx=0 was set at the top of the gap rather than in its

middle. Fig. 19 compares the band structure at kx=0 for

the two scenarios, Δmax=V (Fig. 19a) and Δmax=2V (Fig.

19b). In the second case, qcdw is chosen a little longer to

allow the main band and its translated to cross at –V. As

cdw

q

shown in Fig. 19c and 19d, the gap observed at the

Fermi level by ARPES as a function of kx would be very

similar, although V would be different by a factor two.

One important difference between these two

scenarios concerns the conservation of the number of

occupied states. For the common case of a

homogeneously gapped FS, Ef has to be in the middle of

the gap to conserve the number of electrons. However,

the situation is quite different here, because of the

imperfect nesting. As soon as the crossing between the

original band and its translated band takes place away

from the Fermi level, there is a different number of

occupied states in the gapped and metallic state. In the

case of Fig. 19b, for example, the states above the

crossing position (this region is delimited by black

arrows) were occupied in the metallic state, but are

empty in the CDW state (the weight in the shadow band

is just shifted from the main band, so that this does not

change the counting of occupied states). This roughly

corresponds to a loss of V/Vf*n(k) states, n(k) being the

density of k states and Vf the Fermi velocity. More

generally, the imbalance is just proportional to the

position of the crossing in energy. It is easy to see in Fig.

19d that the loss of states near kx=0 will be compensated

by a gain of states for higher kx. On the contrary, in Fig.

19c, there is no such compensation and there would be a

significant excess of electrons in the CDW state, which

is obviously not self-consistent and requires a shift of Ef.

Direct calculations of the density of states in metallic

and gapped states (see part III.F) shows that the Fermi

level moves very close to the position of scenario (b) to

conserve the number of electrons. This also maximizes

Fig. 19 : (a) and (b) : Black line is the TB band at kx=0, dotted lines

are translated by ±q, with qa=0.68c* and qb=0.733c*. V is chosen to

get the same gap in the two cases : Va=0.4eV and Vb=0.2eV. (c) and

(d) : Thick line is the position of the crossing between the original and

translated bands as a function of kx, for the values of q and V

corresponding to (a) and (b). Thin lines are translated by ±V. Dotted

areas represent the gapped regions.

Fig. 18: (a) Dispersion along the outer FS in LaTe3 for three kz

values (hν=35eV). The TB px band is shown as thick red line and

CDW shadow bands as thin lines. (b) Detail of the dispersion in

LaTe3 at kz=0.2c*. (c) Equivalent situation for YTe3 at kz=0.32c*

(Fig. 10b).

Page 13

13

the gain of electronic energy for a given value of V. This

gives a natural explanation for the choice of the situation

Δmax≈2V observed in Fig. 18.

In principle, probes that also measure unoccupied

states, like STM, could directly confirm or infirm this

scenario. However, they are not k-resolved and, as there

is a large distribution of gaps, the analysis is not

straightforward.18

C. Value of qcdw

Another consequence of the scenario (b) is that qcdw

should change with V. This is an interesting point as a

change of qcdw is indeed observed experimentally. In

scenario (a), qcdw is fixed by the size of the FS to

2kf=0.68c* at kx=0 (in the 3D BZ, qcdw=1-2kf=0.32c*).

As seen in part II.A., this size should not and does not

change with R. On the contrary, in scenario (b) qcdw is

chosen for the crossing to take place at –V for kx=0 and

therefore systematically changes with V.

Satellite positions were observed by TEM13 to

increase from qcdw =0.27c* (for c≈4.4Å) to 0.31c* (for

c≈4.3Å). Our recent x-ray measurements measured

precisely satellites at

(a=4.312Å).17 We report in Fig. 20 values we have

measured for different rare-earth. 17 To obtain a crossing

at E=–V for kx=0, one gets from the TB model :

⎛

=−

cdw

t

2

π

Fig. 20 illustrates the changes of qcdw expected in this

scenario for the two extreme values of tpara determined in

part II. The change with tpara is negligible in front of that

with V. The agreement with the experimental data is

quite spectacular, although the absolute values are

qcdw=0.296c* in TbTe3

()⎟⎟

⎠

⎞

⎜⎜

⎝

+

−

perppara

f

t

EV

Arcq

cos

2

1

slightly larger. This supports this model as the basic

origin of the variation of qcdw. Indeed, there is no

variation of qcdw with R expected from our study of the

nesting properties of the FS in RTe3. If small variations

exist as a function of R, they have to be restricted to the

error bar of tperp or to subtle changes in bilayer splitting,

perpendicular couplings, or interaction between px and pz

(arising for example

orthorhombicities). Such effects are probably important

for a complete description of the CDW. The existence of

two successive transitions

compounds17 or the deviation of YTe3 from the general

trend of RTe3 (Fig. 14) are proofs that there are

subtleties not captured by the TB model. However, they

are probably not able to overcome the strong trend

exposed in Fig. 20.

D. k-dependence of the gap

We have seen that a variation of the gap along the

FS is a natural consequence of the nesting driven CDW,

when the FS exhibits imperfect nesting, even when the

interaction V itself is isotropic. Conversely, the

distribution of the gap over the FS directly informs about

the direction of the nesting. For example, the CDW wave

vector has to be parallel to c* to explain the opening of

the same gap on the square and outer FS at one kx value

(Fig. 13). Such a direction is fully consistent with x-ray

and TEM studies.13,16,17 We have also already seen that

the experimental values of qcdw closely correspond to the

best FS nesting wave vector.

We now study the k-dependence of the gap

quantitatively. Fig. 21a gives examples of leading edge

from slightly different

in heavy rare-earth

Fig. 20 : Red points : experimental values of qcdw measured by x-ray

(ref 17 and 16 for CeTe3), as a function of V=Δ/2 (Δ is taken from

Fig. 14). Black lines : Theoretical variation for qcdw as a function of

V (see text), for tpara=-1.7eV and tpara=-1.85eV.

Fig. 21 : (a) Leading edge spectra in SmTe3 at different kxvalues

and hν=55eV on square part (red) and outer part (blue). The

black line is a fitted Fermi step. (b) EDC spectra on the outer part

in LaTe3 at 35eV and kz=0.35c*.

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14

spectra in SmTe3 at different kx values used to determine

the gap in Fig.13. The Fermi edge (black line on Fig. 21a

with a width of 30meV, corresponding to the

experimental resolution) is very clear, making the

determination of the metallic zones unambiguous. When

the gap opens, the Fermi edge is preserved, but a double

step is commonly found, especially on the outer part. We

believe that, rather than a distribution of gap values, this

is due to the bilayer splitting. Fig. 21b shows data

acquired at 35eV instead of 55eV in Fig. 21a (although

in LaTe3 instead of SmTe3). The structure near Ef

resembles a double peak, with two different gaps, rather

than a line with shoulder. This could be explained by the

variation of intensity of the bilayer split bands with

photon intensity described in II.C.2. In all cases, we have

aligned the Fermi edge on the more deeply gapped part.

We note that slightly different gap values were reported

for the two parts at a same kx value in ref. 19. This could

be due to a different treatment of this bilayer splitting or

to a lower accuracy of this first measurement done only

in 1st BZ, where the intensity of the outer part is very

weak. In all investigated cases, we have observed

identical gaps on the two parts, within experimental

accuracy.

Knowing the value of the CDW wave vector, it is

straightforward to deduce the k-dependence of the gap

expected within the TB model from the variation of

nesting. The principle is that of Fig. 19d and can be

calculated for any value of V. We report on Fig. 13 these

variations as a black line, with values for qcdw taken from

the formula of III.C. The agreement is very satisfying,

although the measured gap seems to fall to zero a little

faster in the measurement than in the theory. A natural

consequence of this simple model is that the position kx

at which the gap becomes zero, depends on the

maximum gap value. We calculate a linear variation of

kx

0

0 for a gap between 0.2 and 0.4eV, which is shown in

Fig. 22 and fits well with the experimental values,

despite some scattering in data points. We conclude that

the general behavior of the gap opening directly results

from the nesting properties of the FS.

E. Metallic parts of the electronic structure

A unique feature of RTe3 is the possibility of

directly observing by ARPES the reconstructed FS, as

we have discussed in ref. 20 and will develop here.

Whereas shadow bands are commonly seen in the

gapped state8,12 (like in Fig.16), they are rarely

detectable in metallic situations (like kx=0.5 in Fig. 15b),

because their intensity when they re-appear at Ef is

already very small. However, observing these lines gives

detailed information about the deviation from nesting

and allow to fully characterize the CDW state.

In Fig. 9a, shadow bands were visible within the

metallic part of the square, although they are rather

weak. Fig. 23 gives two examples of shadow bands near

the junction between square and outer parts (see position

in Fig. 8), for CeTe3 and TbTe3, i.e. for large and small

V. Six bands cross the Fermi level in this area, px and pz,

their 2 shadow bands and their 2 folded bands. The

dotted red and blue lines in Fig. 23a correspond to the

dispersion of px and pz in the CDW state (i.e. including

the shadow band). Note that for TbTe3, the asymmetry in

kz is due to a 7° misalignment of the sample. Although

weak, the shadow bands are clearly detected. MDC cuts

near Ef (Fig.23c and 23d) allow us to estimate their

intensity to 12% of the main line in CeTe3 and 3% in

TbTe3. Theoretically, one expects in this region 8% at

V=0.2eV (CeTe3) and 3% for V=0.14eV (TbTe3), in

reasonable agreement with our experimental estimation.

Fig. 22 : kx value at which the gap on FS becomes zero as a function

of the maximum gap value. Samples are the same than in Fig. 14.

Straight line is the position of kx

value in the TB model. Right axis indicates the corresponding

fraction of the FS contour that remains metallic.

0 calculated as a function of the gap

Fig. 23 : (a) Dispersion in CeTe3 along c* at kx=0.38a* and

hν=55eV. Dotted red and blue lines are dispersion for px and pzin

the CDW state. (b) Dispersion in TbTe3 along a direction at 7° with

respect to c*, at kx=0.35a* and hν=55eV. (c) and (d) MDC cut near

Efcorresponding to (a) and (b). In (c), the red line is a 3 lorentzians

fit.

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15

Note that these predicted intensities are still quite small,

so that it is clear that this work would not be possible

without the large gap in RTe3. As shown in ref. 20, the

evolution of the shadow bands can be followed from

kx=0.3a* up to 0.45a*, making their assignment

unambiguous.

The real shape of the FS can be traced from the

crossing position of these main and shadow bands. It is

more complicated than in Fig. 17, because of the

additional interactions between px and pz and with the

folded FS. The interaction between px and pz is visible in

Fig. 23a, it opens a gap of about 2Vpx-pz≈0.3eV at their

crossing, which replaces the linear dispersions near Ef by

a shallow parabola. This is in good agreement with the

band calculation (see for example Fig. 2, where 2Vpx-

pz≈0.2eV). Taking this interaction into account, the FS

evolves from the 2D TB FS of Fig. 4a to the square and

outer sheets of Fig. 4b (for clarity, a large Vpx-pz≈0.3eV

is used in this figure). The 3D FS is obtained by folding

the 2D with respect to the 3D BZ, as sketched on Fig. 4c.

The main and folded bands also interact. This can be

seen in Fig. 2b, where a gap 2V3D≈0.18eV opens at their

crossing, which takes place at –0.8eV. This is a similar

strength as for the CDW interaction and, indeed, in Fig.

23, folded bands appear with similar intensity as the

CDW shadow bands. After interaction (Fig. 24a), the

outer part breaks into a small oval pocket near the zone

boundary and a larger squared feature. The oval pocket

is clearly present in the experimental data (see Fig. 24c).

The periodicity is the one of the 3D BZ, but the

distribution of the spectral weight is reminiscent of the

2D FS.20 Once again, we take the size of the markers

proportional to the spectral weight.

In Fig. 24, we proceed to the full reconstruction of

the FS in the CDW state. This is similar to Fig. 17 but

we now use the real FS of Fig. 24a instead of that of Fig.

4a. The main effect is a gapping of a large stripe along kz

for about –0.25a*<kx<0.25a*. The remaining fraction of

the square is "closed" at the bottom by the shadow FS.

The structure of the top of the square is more

complicated. In the case of CeTe3 (Fig. 24c), it clearly

does not close but smoothly connects to the shadow FS.

However, this shape sensitively depends on the relative

strength of Vpx-pz and Vq, and also probably on the

bilayer splitting. For YTe3 (Fig. 5), the top part of the

square is clearly closed. The fact that one can sort out

these details is mainly the consequence of much

improved resolution and data rate in modern ARPES. To

close the other side of the pocket, along the outer part,

the mechanism is similar to the one band case of Fig. 17.

One may wonder if the interaction at qcdw is

equivalent to that at (c*-qcdw), as we have seen that it is

only when the weak folded FS is considered that qcdw

becomes meaningful. If it is, we should observe in our

data a gap at the crossing between the folded and CDW

shadow bands. On the contrary, it seems in Fig. 23 that

these two bands cross without interacting. In Fig. 24b,

we calculate the FS in two extreme cases, with

V(qcdw)=0 (kx>0) and V(qcdw)= V(c*-qcdw) (kx<0). The

crossing between folded and shadow FS is indicated by

black arrows and this region is quite different in the two

cases. The comparison with the experimental map of Fig.

24c clearly favors the first case.

This is a particularity of the CDW in this system,

which is dominated by the in-plane coupling and is in

fact essentially 2D. If not properly recognized, this could

mimic a deviation from a sinusoidal distortion. In ref. 16,

it was proposed that the CDW is commensurate within

discommensurate domains, in order to create patterns of

Fig. 24 : (a) 3D FS including interaction between px and pz and between main and folded bands (bilayer splitting is omitted). The green square is

the 3D BZ. The size of the markers is proportional to the spectral weight. The red arrows correspond to CDW wave vectors (c*-qcdw) and qcdw. (b)

Red markers : weight of the reconstructed FS calculated within the TB model with V(qcdw)=0 at kx>0, and V(qcdw)= V(c*-qcdw) at kx<0. Black

contours are guide for the eyes of square and outer parts, dotted contours are for folded parts. (c) Zoom on the FS pockets measured in CeTe3with

a photon energy of 55eV.

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- Available from Zahid Hussain · May 22, 2014
- Available from arxiv.org