Evidence for an excitonic insulator phase in 1T-TiSe$_{2}$
APS/123-QED
Evidence for an excitonic insulator phase in 1T -TiSe2
H. Cercellier,∗ C. Monney, F. Clerc, C. Battaglia, L. Despont, M. G. Garnier, H. Beck, and P. Aebi
Institut de Physique, Université de Neuchâtel, CH-2000 Neuchâtel, Switzerland
L. Patthey
Swiss Light Source, Paul Scherrer Institute, CH-5232 Villigen, Switzerland
H. Berger
Institut de Physique de la Matière Complexe, EPFL, CH-1015 Lausanne, Switzerland
(Dated: October 22, 2018)
We present a new high-resolution angle-resolved photoemission study of 1T -TiSe2 in both, its
room-temperature, normal phase and its low-temperature, charge-density wave phase. At low tem-
perature the photoemission spectra are strongly modified, with large band renormalisations at high-
symmetry points of the Brillouin zone and a very large transfer of spectral weight to backfolded
bands. A theoretical calculation of the spectral function for an excitonic insulator phase reproduces
the experimental features with very good agreement. This gives strong evidence in favour of the
excitonic insulator scenario as a driving force for the charge-density wave transition in 1T -TiSe2.
PACS numbers:
Transition-metal dichalcogenides (TMDC’s) are lay-
ered compounds exhibiting a variety of interesting phys-
ical properties, mainly due to their reduced dimension-
ality [1]. One of the most frequent characteristics is a
ground state exhibiting a charge-density wave (CDW),
with its origin arising from a particular topology of the
Fermi surface and/or a strong electron-phonon coupling
[2]. Among the TMDC’s 1T -TiSe2 shows a commensu-
rate 2×2×2 structural distortion below 202 K, accom-
panied by the softening of a zone boundary phonon and
with changes in the transport properties [3, 4]. In spite
of many experimental and theoretical studies, the driv-
ing force for the transition remains controversial. Sev-
eral angle-resolved photoelectron spectroscopy (ARPES)
studies suggested either the onset of an excitonic insula-
tor phase [5, 6] or a band Jahn-Teller effect [7]. Further-
more, TiSe2 has recently attracted strong interest due to
the observation of superconductivity when intercalated
with Cu [8]. In systems showing exotic properties, such
as Kondo systems for example [9], the calculation of the
spectral function has often been a necessary and deci-
sive step for the interpretation of the ARPES data and
the determination of the ground state of the systems. In
the case of 1T -TiSe2, such a calculation for an excitonic
insulator phase lacked so far.
In this letter we present a high-resolution ARPES
study of 1T -TiSe2, together with theoretical calculations
of the excitonic insulator phase spectral function for this
compound. We find that the experimental ARPES spec-
tra show strong band renormalisations with a very large
transfer of spectral weight into backfolded bands in the
low-temperature phase. The spectral function calculated
for the excitonic insulator phase is in strikingly good
Electronic address: herve.cercellier@unine.ch
agreement with the experiments, giving strong evidence
for the excitonic origin of the transition.
The excitonic insulator model was first introduced in
the sixties, for a semi-conductor or a semi-metal with a
very small indirect gap EG [10, 11, 12, 13]. Thermal ex-
citations lead to the formation of holes in the valence
band and electrons in the conduction band. For low
free carrier densities, the weak screening of the electron-
hole Coulomb interaction leads to the formation of sta-
ble electron-hole bound states, called excitons. If the
exciton binding energy EB is larger than the gap energy
EG, the system becomes unstable upon formation of exci-
tons. This instability can drive a transition to a coherent
ground state of condensed excitons, with a periodicity
given by the spanning vector w that connects the va-
lence band maximum to the conduction band minimum.
In the particular case of TiSe2, there are three vectors
(wi, i = 1, 2, 3) connecting the Se 4p-derived valence
band maximum at the Γ point to the three symmetry-
equivalent Ti 3d-derived conduction band minima at the
L points of the Brillouin zone (BZ) (see inset of fig. 1b)).
Our calculations are based on the BCS-like model of
Jérome, Rice and Kohn [12], adapted for multiple wi.
The band dispersions for the normal phase have been
chosen of the form
ǫv(k) = ǫ
v + ~
k2x + k
+ tv cos(
c(k,wi) = ǫ
c + ~
( (kx − wix)
(ky − wiy)
+tc cos
(2π(kz − wiz)
for the valence (ǫv) and the three conduction (ǫ
c) bands
respectively, with c the lattice parameter perpendicular
to the surface in the normal (1×1×1) phase, tv and tc the
amplitudes of the respective dispersions perpendicular to
the surface and mv, mc the effective masses.
http://arxiv.org/abs/0704.0159v1
mailto:herve.cercellier@unine.ch
The parameters for equations 1 were derived from pho-
ton energy dependent ARPES measurements carried out
at the Swiss Light Source on the SIS beamline, using
a Scienta SES-2002 spectrometer with an overall energy
resolution better than 10 meV, and an angular resolution
better than 0.5◦. The fit to the data gives for the Se 4p
valence band maximum -20 ± 10 meV, and for the Ti 3d
conduction band a minimum -40 ± 5 meV [14]. From our
measurements we then find a semimetallic band structure
with a negative gap (i.e. an overlap) EG=-20 ± 15 meV
for the normal phase of TiSe2, in agreement with the lit-
erature [15]. The dispersions deduced from the ARPES
data are shown in fig. 1a) (dashed lines).
Within this model the one-electron Green’s functions
of the valence and the conduction bands were calculated
for the excitonic insulator phase. For the valence band,
one obtains
Gv(k, z) =
z − ǫv(k)−
|∆|2(k,wi)
z − ǫc(k+wi)
. (2)
This is a generalized form of the equations of Ref. [12]
for an arbitrary number of wi. The order parameter ∆ is
related to the number of excitons in the condensed state
at a given temperature. For the conduction band, there
is a system of equations describing the Green’s functions
Gic corresponding to each spanning vector vector wi:
z − ǫic(k+wi)
c(k+wi, z) = 1 +∆
∗(k,wi)
∆(k,wj)G
c(k+wj, z)
z − ǫv(k)
This model and the derivation of the Green’s functions
will be further described elsewhere [16].
The spectral function calculated along several high-
symmetry directions of the BZ is shown in fig. 1a) for an
order parameter ∆=0.05 eV. Its value has been chosen for
best agreement with experiment. The color scale shows
the spectral weight carried by each band. For presen-
tation purposes the δ-like peaks of the spectral function
have been broadened by adding a constant 30 meV imagi-
nary part to the self-energy. In the normal phase (dashed
lines), as previously described we consider a semimetal
with a 20 meV overlap, with bands carrying unity spec-
tral weight. In the excitonic phase, the band structure
is strongly modified. The first observation is the appear-
ance of new bands (labeled C1, V2 and C3), backfolded
with the spanning vector w = ΓL. The C1, V2 and C3
branches are the backfolded replicas of branches C2, V3
and C4 respectively. In this new phase the Γ and L points
are now equivalent, which means that the excitonic state
has a 2×2×2 periodicity of purely electronic origin, as ex-
pected from theoretical considerations [10, 12]. Another
effect of exciton condensation is the opening of a gap in
the excitation spectrum. This results in a flattening of
the valence band near Γ in the ΓM direction (V1 branch)
and in the AΓ direction (V3 branch), and also an upward
bend of the conduction band near L and M (C2 and C4
Γ Μ Α ΓL
FIG. 1: : a) Spectral function of the excitonic insulator in a
1T structure calculated for a 20 meV overlap and an order
parameter ∆=0.05 eV. The V1-V3 (resp. C1-C4) branches
refer to the valence (resp. conduction) band. Dashed lines
correspond to the normal phase (∆=0). The path in recipro-
cal space is shown in red in the inset. b) Spectral weight of
the different bands. Inset : bulk Brillouin zone of 1T -TiSe2.
branches). It is interesting to notice that in the vicinity of
these two points, the conduction band is split (arrows).
This results from the backfolding of the L points onto
each other, according to the new periodicity of the exci-
tonic state [17]. The spectral weight carried by the bands
is shown in fig. 1b). The largest variations occur near
the Γ, L and M points, where the band extrema in the
normal phase are close enough for excitons to be created.
Away from these points, the spectral weight decreases in
the backfolded bands (C1, V2, C3) and increases in the
others. The intensity of the V1 branch, for example, de-
creases by a factor of 2 when approaching Γ, whereas
the backfolded C1 branch shows the opposite behaviour.
Such a large transfer of spectral weight into the back-
folded bands is a very uncommon and striking feature.
Indeed, in most compounds with competing potentials
(CDW systems, vicinal surfaces,...), the backfolded bands
carry an extremely small spectral weight [18, 19, 20]. In
these systems the backfolding results mainly from the in-
fluence of the modified lattice on the electron gas, and
the weight transfer is related to the strength of the new
crystal potential component. Here, the case of the exci-
tonic insulator is completely different, as the backfolding
is an intrinsic property of the excitonic state. The large
Τ=250 Κ Τ=250 Κ
Τ=65 Κ Τ=65 Κ
FIG. 2: : ARPES spectra of 1T -TiSe2 for a) the normal and b)
the low temperature phase. Thick dotted lines are parabolic
fits to the bands in the normal phase and thin dotted lines
are guides to the eye for the CDW phase. Fine lines follow
the dispersion of the 4p sidebands (see text).
transfer of spectral weight is then a purely electronic ef-
fect, and turns out to be a characteristic feature of the
excitonic insulator phase.
Fig. 2 shows ARPES spectra recorded at a photon
energy hν=31 eV as a function of temperature. At this
photon energy, the normal emission spectra correspond
to states located close to the Γ point. For the sake of sim-
plicity the description is in terms of the surface BZ high-
symmetry points Γ̄ and M̄ . The 250 K spectra exhibit
the three Se 4p-derived bands at Γ̄ and the Ti 3d-derived
band at M̄ widely described in the literature [5, 6, 7].
The thick dotted lines (white) are fits by equation 1, giv-
ing for the topmost 4p band a maximum energy of -20
± 10 meV, and for the Ti 3d a minimum energy of -40
± 5 meV. The small overlap EG=-20 ± 15 meV in the
normal phase is consistent with the excitonic insulator
scenario, as the exciton binding energy is expected to be
close to that value. [5, 6]. The position of both band
maxima in the occupied states is most probably due to
a slight Ti overdoping of our samples [3]. In our case, a
transition temperature of 180 ± 10 K was found from dif-
ferent ARPES and scanning tunneling microscopy mea-
surements, indicating a Ti doping of less than 1 %. On
the 250 K spectrum at Γ̄, the intensity is low near normal
emission. This reduced intensity and the residual inten-
sity at M̄ around 150 meV binding energy (arrows) may
arise from exciton fluctuations (see reduction of spectral
Τ=250 Κ Τ=250 Κ
Τ=65 Κ Τ=65 Κ
∆=0.05 eV
FIG. 3: : Theoretical spectral function of 1T -TiSe2, calcu-
lated along the path given by the free electron final state
approximation shown in the inset. a) normal state and b)
low temperature phase (see text).
weight near Γ in the V1 branch in fig. 1b). Matrix ele-
ments do not appear to play a role as the intensity vari-
ation only depends very slightly on photon energy and
polarization. In the 65 K data (fig. 2b)), the topmost
4p band flattens near Γ̄ and shifts to higher binding en-
ergy by about 100 meV (thin white, dotted line). This
shift is accompanied by a larger decrease of the spectral
weight near the top of the band. The two other bands
(fine black lines) are only slightly shifted and do not ap-
pear to participate in the transition. In the M̄ spectrum
strong backfolded valence bands can be seen, and the
conduction band bends upwards, leading to a maximum
intensity located about 0.25 Å−1 from M̄ (thin white dot-
ted line). This observation is in agreement with Kidd et
al. [6], although in their case the conduction band was
unoccupied in the normal phase.
The calculated spectral functions corresponding to the
data of fig. 2 are shown in fig. 3, using the free-electron fi-
nal state approximation with a 10 eV inner potential and
a 4.6 eV work function (see inset). The effect of tempera-
ture was taken into account via the order parameter and
the Fermi function. Only the topmost valence band was
considered, as the other two are practically not influenced
by the transition (see above, fig. 2). The behavior of this
band is extremely well reproduced by the calculation. In
the 65 K calculation the valence band is flattened near Γ̄,
and the spectral weight at this point is reduced to 44 %,
close to the experimental value of 35 %. The agreement
E-EF (eV)Model ARPES
−0.025
FIG. 4: : Near-EF constant energy cuts in the vicinity of the
Γ point. The theoretical data correspond to fig. 3b and the
ARPES data are taken from the low-temperature data of fig.
is very satisfying, considering that the calculation takes
into account only the lowest excitonic state. The exper-
imental features appear broader than in the calculation,
but at finite temperatures one may expect the existence
of excitons with non-zero momentum, leading to a spread
of spectral weight away from the high-symmetry points.
In the near-M̄ spectral function, the backfolded va-
lence band is strongly present in the 65 K calculation,
with comparable spectral weight as at Γ̄ and as the con-
duction band at M̄ . The conduction band maximum
intensity is located away from M̄ as in the experiment.
The small perpendicular dispersion of the free-electron
final state causes an asymmetry of the intensity of the
conduction band on each side of M̄ , which is also visible
in fig. 2. In our calculation, as opposed to the ARPES
spectra, the conduction band is unoccupied and only the
occupied tail of the peaks is visible. This difference may
be simply due to the final state approximation used in
the calculation, a slight shift of the chemical potential
due to the transition, or to atomic displacements that
would shift the conduction band [6, 7, 21]. Such atomic
displacements, in terms of a band Jahn-Teller effect, were
suggested as a driving force for the transition. However,
the key point is that, although the lattice distortion may
shift the conduction band, the very small atomic displace-
ments (≈ 0.02 Å [3]) in 1T -TiSe2 are expected to lead
to a negligable spectral weight in the backfolded bands
[20]. As an example, 1T -TaS2, another CDW compound
known for very large atomic displacements [22] (of or-
der > 0.1 Å) introduces hardly detectable backfolding of
spectral weight in ARPES. Clearly, an electronic origin
is necessary for obtaining such strong backfolding in the
presence of such small atomic displacements. Therefore,
our results allow to rule out a Jahn-Teller effect as the
driving force for the transition of TiSe2.
Furthermore, the ARPES spectra also show evidence
for the backfolded conduction band at the Γ̄ point. Fig.
4 shows constant energy cuts around the Fermi energy,
taken from the data of fig. 2b and 3b (arrows). In the
ARPES data two slightly dispersive peaks, reproduced
in the calculation, clearly cross the Fermi level. These
features turn out to be the populated tail of the back-
folded conduction band, whose centroid is located just
above the Fermi level. To our knowledge no evidence
for the backfolding of the conduction band had been put
forward so far.
In summary, by comparing ARPES spectra of 1T -
TiSe2 to theoretical predictions for an excitonic insula-
tor, we have shown that the superperiodicity of the ex-
citonic state with respect to the lattice results in a very
large transfer of spectral weight into backfolded bands.
This effect, clearly evidenced by photoemission, turns
out to be a characteristic feature of the excitonic insula-
tor phase, thus giving strong evidence for the existence
of this phase in 1T -TiSe2 and its prominent role in the
CDW transition.
Skillfull technical assistance was provided by the work-
shop and electric engineering team. This work was sup-
ported by the Fonds National Suisse pour la Recherche
Scientifique through Div. II and MaNEP.
[1] J. A. Wilson et al., Adv. Phys. 24, 117 (1975).
[2] F. Clerc et al., Phys. Rev. B 74, 155114 (2006).
[3] F. J. Di Salvo et al., Phys. Rev. B 14, 4321 (1976).
[4] M. Holt et al., Phys. Rev. Lett. 86, 3799 (2001).
[5] T. Pillo et al., Phys. Rev. B 61, 16213 (2000).
[6] T. E. Kidd et al., Phys. Rev. Lett. 88, 226402 (2002).
[7] K. Rossnagel et al., Phys. Rev. B 65, 235101 (2002).
[8] E. Morosan et al., Nature Physics 2, 544 (2006).
[9] D. Malterre et al., Adv. Phys. 45, 299 (1996).
[10] W. Kohn, Phys. Rev. Lett. 19, 439 (1967).
[11] B. I. Halperin and T. M. Rice, Rev. Mod. Phys. 40, 755
(1968).
[12] D. Jérome et al., Phys. Rev. 158, 462 (1967).
[13] F. X. Bronold and H. Fehske, Phys. Rev. B 74, 165107
(2006).
[14] The fit parameters are : ǫ0v=-0.08±0.005 eV, mv=-
0.23±0.02 me, where me is the free electron mass,
tv=0.06±0.005 eV ; ǫ
c=-0.01±0.0025 eV, m
c=5.5±0.2
me, m
c=2.2±0.1 me, tc=0.03±0.0025 eV
[15] O. Anderson et al., Phys. Rev. Lett. 55, 2188 (1985).
[16] C. Monney et al., to be published
[17] J. A. Wilson et al., Phys. Rev. B 18, 2866 (1978).
[18] C. Didiot et al., Phys. Rev. B 74, 081404(R) (2006).
[19] C. Battaglia et al., Phys. Rev. B 72, 195114 (2005).
[20] J. Voit et al., Science 290, 501 (2000).
[21] M. H. Whangbo and E. Canadell, J. Am. Chem. Soc.
114, 9587 (1992).
[22] A. Spijkerman et al., Phys. Rev. B 56, 13757 (1997).
