arXiv:cond-mat/0612064v3 [cond-mat.str-el] 31 Mar 2007
A novel mechanism of charge density wave in a transition metal dichalcogenide
D. W. Shen1, B. P. Xie1, J. F. Zhao1, L. X. Yang1, L. Fang2,
J. Shi3, R. H. He4, D. H. Lu4, H. H. Wen2, and D.L. Feng1∗
1Department of Physics, Applied Surface Physics State Key Laboratory, Fudan University, Shanghai 200433, China
2National Lab for Superconductivity, Institute of Physics and National Lab for Condensed Matter Physics,
Chinese Academy of Sciences, P.O. Box 603, Beijing 100080, P. R. China
3School of Physics, Wuhan University, Wuhan, 430072, P. R. China and
4Department of Applied Physics and Stanford Synchrotron Radiation Laboratory, Stanford University, Stanford, CA 94305, USA
(Dated: February 6, 2008)
Charge density wave, or CDW, is usually associated with Fermi surfaces nesting. We here report
a new CDW mechanism discovered in a 2H-structured transition metal dichalcogenide, where the
two essential ingredients of CDW are realized in very anomalous ways due to the strong-coupling
nature of the electronic structure. Namely, the CDW gap is only partially open, and charge density
wavevector match is fulfilled through participation of states of the large Fermi patch, while the
straight FS sections have secondary or negligible contributions.
PACS numbers: 71.18.+y, 71.45.Lr, 79.60.-i
It has been a standard textbook-example that charge
density wave (CDW), one of the main forms of ordering
in solid, is mostly associated with nesting Fermi surface
(FS) sections. In charge ordered materials ranging from
one-dimensional (1D) (TaSe4)2I and blue bronze[1, 2]
to two-dimensional (2D) manganites, and from sur-
face reconstruction in weak correlated metals to checker
board pattern of strongly correlated high temperature
superconductors[3, 4], the charge fluctuations associated
with the ordering wave vector scatter the electrons be-
tween two nested FS sections and effectively drive the
system into an ordered ground state. However, this clas-
sical picture failed in the very first 2D CDW compound
discovered in 1974, i.e. the transition metal dichalco-
genides (TMD’s). The 2H-structured TMD’s have a
hexagonal lattice structure, and in its CDW phase, a
3 × 3 superlattice forms[5, 6]. It was found that the or-
dering wavevectors do not match the nested FS sections,
and generally no CDW energy gap was observed at the
FS[7, 8]. After decades of continuous effort, the origin
of CDW for the 2H-structured TMD’s has been a long
standing mystery. As a result, the subtle details of the
competition and coexistence of CDW and superconduc-
tivity in TMD’s remain to be revealed.
In this letter, we studied the electronic origin of the
CDW in a 2H-TMD, 2H-NaxTaS2, by angle-resolved
photoemission spectroscopy (ARPES). The CDW mech-
anism in this material was discovered after the revelation
of the following exotic properties. i) The electronic struc-
ture exhibits strong coupling nature, with finite density
of states at the Fermi energy(EF) over almost the entire
Brillouin zone(BZ), forming so-called Fermi patches. ii)
In the CDW state, only a fraction of the states at EF is
gapped. iii) The density of states near EF directly corre-
lates with the ordering strength. iv) Fermi patch, instead
of Fermi surface, is relevant for CDW. We show that this
new “Fermi-patch mechanism” for CDW is rooted in the
strong coupling nature of the electronic structure and it
may be a general theme of ordering in the strong coupling
regime of various models, and applicable to systems with
similar electronic structure.
For the systematic studies of the electronic structure
in a 2H-TMD compound, 2H-NaxTaS2with 2%, 5% and
10% Na concentration were synthesized with CDW tran-
sition temperature TCDW’s are 68K, 65K and 0K respec-
tively. The samples are labelled as CDW68K, CDW65K,
and CDW0Khereafter. The corresponding supercon-
ducting transition temperatures are 1.0K, 2.3K and 4.0K
respectively, a manifestation of the competition between
CDW and superconductivity in this system. The data
were mainly collected using 21.2eV Helium-I line of a
discharge lamp combined with a Scienta R4000 analyzer,
and partial measurements were carried out on beam line
5-4 of SSRL. The overall energy resolution is 8 meV , and
the angular resolution is 0.3 degrees.
(CDW65K) and without CDW (CDW0K) are compared
in Fig.1a-d. In both cases, the spectral lineshapes are
remarkably broad, and no quasiparticle peaks in the
conventional sense are observed.
is of the same order as the dispersion, which clearly
indicates the incoherent nature of the spectrum and
the system is in the strong coupling regime.
the normalized spectra at M as reference, spectra
at Γ, K, and the Fermi crossing of the Γ-M cut are
compared in Fig.1e. The difference between CDW0K
and CDW65Kis striking.
ilar density of states at the Fermi crossing area, for
momentum regions away from the FS, CDW65Khas
much stronger spectral weight than CDW0K, no matter
whether it is inside the occupied region (M point),
or in the unoccupied region (Γ and K points) in the
band structure calculations. Particularly, one finds
that even when the spectral centroid is well below EF,
spectra taken onsampleswith
The large linewidth
Although they have sim-
FIG. 1: Typical ARPES spectra for a-b, CDW0Kat T=15K and c-d, CDW65Kat T=95K respectively along the marked cuts
in the Brillouin zone in panels f and g. e, Comparison of photoemission spectra for different 2H-NaxTaS2’s at high symmetry
points M, Γ, K and at the Fermi crossing of cut1. f and g, The photoemission intensity integrated within 10meV of EF is
shown for CDW0Kand CDW65Krespectively. (Both data were taken at 15K and the image was 6-fold symmetrized.) The
Fermi surfaces are marked by dashed lines, where the antibonding and bonding bands could be resolved for the Γ pockets in
the momentum distribution curve (one example is shown in h for CDW0K).
the finite residual weight at EF beyond background
exists around M. We note that the CDW0Ksample has
higher Na doping than CDW65K, yet its lineshape is
generally sharper. Therefore, disorder effects induced by
the dopants should be negligible. The residual weight
observed near EF thus should be associated with the
intrinsic strong coupling nature of the system, which
(within several kBTCDW around EF) shows a monotonic
correlation with TCDW in Fig.1i.
Although the spectra are broad, Fermi surfaces could
still be defined as the local maximum of the spectral
weight at EF, which are plotted in Fig.1f-g for
CDW0Kand CDW65Krespectively. Three FS pockets
can be identified through the momentum distribution
curve analysis: two hole pockets around Γ, as exem-
plified in Fig.1h, and one hole pocket around K. Bi-
layer band splitting of the two TaS2layers in a unit cell
manifests itself as the inner and outer Gamma pockets,
while the splitting of the K pockets is indistinguishable.
The FS volumes are evaluated to be 1.05 ± 0.01 and
1.11 ± 0.01 electrons per layer for the 5% and 10% Na
doped NaxTaS2samples respectively, consistent with the
nominal dopant concentrations. This resembles the high
temperature superconductors, where the Fermi surfaces
defined in this conventional way are consistent with the
band structure calculations and follow the Luttinger sum
rule quite well, yet there are large Fermi patches near
the antinodal regions. In CDW65Kcase, almost the
entire BZ appear to be one gigantic Fermi patch. How-
ever, so far, searches for the CDW mechanism are mostly
centered on the Fermi surfaces, not the Fermi patch.
The differences between CDW65Kand CDW0Kspec-
tra in the Fermi patch region(Fig.1e) are intriguing.
Fig.2a-b show spectra taken from M at different tem-
peratures for CDW65Kand CDW0Krespectively. While
the CDW0Kspectra simply exhibit a clear Fermi cross-
ing and thermal broadening, the CDW65Kspectra appear
very anomalous. Take the spectrum at 7K as an exam-
ple, while the upper part of the spectrum is suppressed to
higher binding energies, the middle point of the leading
edge of the lower part still matches EF. There is an ap-
parent turning point between these two parts of the spec-
trum. By dividing the spectra with the corresponding
finite temperature Fermi-Dirac distribution functions, in
Fig.2c, one clearly observes that about 29% of the spec-
tral weight at EF has been suppressed for CDW65K. An
energy gap has opened on part of the states here, which
is estimated to be ∼35meV based on the middle point of
the leading edge. Contrastively, there is no sign of gap
opening for CDW0K(Fig.2d).
It was recently suggested by Barnett and coworkers
that a 2H-TMD system is decoupled into three sub-
lattices. While one of the sub-lattices is undistorted and
gapless below TCDW, the other two are gapped at the
FS. For CDW65K, similar behavior is observed except
that gap does not open at Fermi Surface and just about
1/3 of the spectrum is gapped. Phenomenologically, one
can decompose the spectra into gapped and ungapped
components. The ungapped component is simulated as
Au(k,ω,T) = αA(k,ω,85K)
, where A(k,ω,T) is the spectral function, f(ω,T) is the
resolution convoluted finite temperature Fermi-Dirac dis-
Normalized Intensity (Arb. Units)
CDW gap (meV)
ARPES spectra taken at M for different temperatures for
a, CDW65Kand b, CDW0K. c-d, The spectra in a and b di-
vided by the resolution convoluted Fermi-Dirac distribution at
the corresponding temperatures. e, Each spectrum in a is de-
composed into a gapped component and an ungapped compo-
nent. f, The temperature dependence of the CDW gap. The
solid line is the fit to a mean field formula, ∆0
The CDW gap measurements of 2H-NaxTaS2.
tribution function, and α = 0.71. The gapped compo-
Ag(k,ω,T) = A(k,ω,T) − Au(k,ω,T)
in Fig.2e, where the ungapped components exhibit the
same behavior as in the CDW0Kspectrum, and the
gapped components clearly reveal a clean energy gap
of about 35meV at 7K. We emphasize that although
the decomposition method is adopted hereafter, all the
qualitative results can be obtained through the conven-
tional method in Fig.2c as well. The temperature evo-
lution of the gap is shown in Fig.2f for two different
CDW65Ksamples. Interestingly, the gap does not sat-
urate at low temperature, and it can be fitted very
well to ∆0
form at temperature close to TCDW. Here the fitted
Spectra at other momenta could be decomposed in
the same way quite robustly, and the CDW gap
was mapped over the entire Brillouin zone for CDW65K
(Fig.3a). Strikingly, finite CDW gap exists over most
of the Brillouin zone. Its maximum locates around M,
and no gap is observed around the inner Γ Fermi pocket
and within. Noticeably, the gap is finite in the K Fermi
pockets, as there is finite spectral weight at EF. Close
comparisons of spectra at various momenta are shown
in Fig.3b-d. In between M and the Fermi crossing
(Fig.3b), the upper part of the low temperature spec-
trum is overlayed with the normal state spectrum, which
is a sign of gap opening. In Fig.3c, for spectrum at the
inner Γ Fermi pocket, no sign of gap opening is observed,
consistent with previous studies[7, 8]. Fig.3d illustrates
The decomposition of spectra in Fig.2a is shown
TCDW, which is the mean field theory
Normalized Intensity (Arb. Unit)
-0.4-0.3 -0.2 -0.1
FIG. 3: The gap map over the Brillouin zone. a, The false
color plot of the CDW gap in the first Brillouin zone of
CDW65K, where the dashed lines indicate the Fermi surfaces.
States in the gapped region could be connected by the CDW
wavevectors, Qi (i=1,2,3), as indicated by the double-head
arrows. b-d, comparison of the typical spectra in normal and
CDW state at various momenta marked by the circled let-
ters in a. The gapped spectra are decomposed into ungapped
(dashed lines) and gapped portion (dotted lines).
that a gap of 15meV is observed at the saddle point of
the band calculation. An alternative mechanism was pro-
posed for 2H-TMD’s involving the scattering between
saddle points, which would cause a singularity in density
of states, and thus an anomaly in the dielectric response
function. The gap near this point has been reported
before, but the distance between these points do not
match the CDW ordering wavevectors[22, 23]. In the
current gap map, nothing abnormal is observed for this
The CDW in TMD’s is associated with structural
transitions[5, 6], therefore, electron-phonon interactions
are also crucial in the problem. Since the low energy
electronic structure of 2H-NaxTaS2is dominated by the
Ta 5d electrons , among which Coulomb interactions
are usually weak, the broad ARPES lineshape would
suggests strong electron-phonon interactions.
in the dispersion corresponding to the phonon energy
scale was found for 2H-NaxTaS2 (not shown here) as
in other 2H-TMD compounds. In this context, the
anisotropic gap distribution might be attributed to the
anisotropy of electron-phonon couplings [25, 26], and
states in the ungapped region simply may not couple
with the relevant phonons.
i.e. states in different gapped regions need to be cou-
pled by phonons with the CDW wavevectors, can now be
fulfilled in the gap map, as illustrated by the arrows in
Fig.3a. Qi=ai*/3,(i=1,2,3) here are the CDW wavevec-
tors, ai*’s being the reciprocal lattice vectors along the
three Γ-M directions. This also explains why the size
One critical requirement,
FIG. 4: a Autocorrelation of the ARPES intensity at EF in
the normal state of CDW65Kfor wavevector along the Γ-M
direction. The dashed line is an exponential decay plus a con-
stant, which is deducted in b to highlight the structure. c The
partial autocorrelation in the 2D momentum transfer space
obtained in the same way as in b. Repeated zone scheme
is taken in the integration, and thus C(? q,0) is symmetric in
respect to the boundaries (white hexagon).
of the Fermi surfaces can vary significantly for different
2H-TMD systems, with nearly system-independent CDW
ordering wavevectors[22, 23, 27].
One open question in the above picture is that charge
fluctuations with other wavevectors are also allowed, and
it is not obvious why Qi’s are special. For cuprate su-
perconductors, it has been demonstrated that the auto-
correlation of ARPES spectra,
C(? q,ω) ≡
A(?k,ω)A(?k + ? q,ω)d?k
, could give a reasonable count for the charge modula-
tions observed by STM[28, 29]. This joint density-of-
states describes the phase space for scattering of elec-
trons from the state at?k to the state at?k +? q by certain
modes with wavevector ? q. Therefore, one would expect
that it peaks at the ordering wavevector for ω = 0 near
the phase transition of static order. Since the states in
the gapped Fermi patch are responsible for the CDW
here, autocorrelation analysis is conducted in the nor-
mal state to study the CDW instabilities of CDW65K
over regions that would be gapped below TCDW. The
resulting C(? q,0) is shown in Fig.4a for ? q along the Γ-
M direction, which is mainly consisted of a component
at ? q = 0 that exponentially decays, and several features
in Fig.4b, where a peak is clearly observed around the
CDW ordering wavevector. The peak at ? q = 0 would
require coupling to very long wavelength phonons, which
presumably is very weak. Consistently, a recent calcu-
lation has shown that an optical phonon branch softens
significantly around Qi, and no sign of softening is ob-
served at ? q = 0. In the 2D partial C(? q,0) map (ob-
tained after deducting the exponential decaying part and
the constant background)in Fig.4c, although there are a
few local maxima corresponding to various possible or-
derings, the highest peaks are those at Qi. Therefore,
our results suggest that the electronic structure is in fa-
vor of the charge instability at Qi’s, and eventually the
system becomes unstable to the CDW formation below
TCDW in collaboration with the phonons. Furthermore,
it is also consistent with the positive correlation between
the spectral weight near EF and TCDW in Fig.1i.
The competition and coexistence of CDW and super-
conductivity can be understood within the new frame-
work. Recent photoemission studies have revealed that
superconducting gap opens at the K and Γ pockets[10,
25]. The CDW gap opens at a temperature higher than
the superconducting transition, but it just partially sup-
presses the density of states around the K pocket and
outer Γ pocket.Therefore, as observed in most 2H-
TMD’s, superconductivity is suppressed but not elimi-
nated by the CDW.
To summarize, the Fermi-patch mechanism of CDW
in 2H-NaxTaS2 is characterized by the realization of
both ingredients of the CDW, energy gap and wavevector
match on the Fermi patches. Unlike other CDW mecha-
nisms based on band structure effects, it is rooted in the
strong-coupling nature of its electronic structure, which
provides phase space needed for CDW fluctuations. Fur-
thermore, this mechanism would be realized not only in
polaronic systems, but also in materials where strong
electron correlations could cause Fermi patches, and thus
We thank Profs. H. Q, Lin, J. L. Wang, Q. H. Wang, J.
X. Li, Z. D. Wang and F. C. Zhang for helpful discussions.
This work is supported by NSFC, MOST’s 973 project:
2006CB601002 and 2006CB921300.
∗Electronic address: firstname.lastname@example.org
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