Pseudogap and charge density waves in two dimensions
Pseudogap and charge density waves in two dimensions 
S. V. Borisenko1, A. A. Kordyuk1,2, A. N. Yaresko3, V. B. Zabolotnyy1, D. S. Inosov1, R. Schuster1, 
B. Büchner1, R. Weber4, R. Follath4, L. Patthey5, H. Berger6 
 1Leibniz-Institute for Solid State Research, IFW-Dresden, D-01171, Dresden, Germany 
2Institute of Metal Physics, 03142 Kyiv, Ukraine 
3Max-Planck-Institute for the Physics of Complex Systems, Dresden, Germany 
4BESSY, Berlin, Germany 
5Swiss Light Source, Paul Scherrer Institut, CH-5234 Villigen, Switzerland 
6Institute of Physics of Complex Matter, EPFL, 1015 Lausanne, Switzerland 
An interaction between electrons and lattice vibrations (phonons) results in two fundamental 
quantum phenomena in solids: in three dimensions it can turn a metal into a superconductor 
whereas in one dimension it can turn a metal into an insulator1, 2, 3.  In two dimensions (2D) 
both superconductivity and charge-density waves (CDW)4, 5 are believed to be anomalous. In 
superconducting cuprates, critical transition temperatures are unusually high and the energy 
gap may stay unclosed even above these temperatures (pseudogap). In CDW-bearing 
dichalcogenides the resistivity below the transition can decrease with temperature even faster 
than in the normal phase6, 7 and a basic prerequisite for the CDW, the favourable nesting 
conditions (when some sections of the Fermi surface appear shifted by the same vector), seems 
to be absent8, 9, 10. Notwithstanding the existence of alternatives11, 12, 13, 14, 15 to conventional 
theories1, 2, 3, both phenomena in 2D still remain the most fascinating puzzles in condensed 
matter physics.  Using the latest developments in high-resolution angle-resolved 
photoemission spectroscopy (ARPES) here we show  that the normal-state pseudogap also 
exists in one of the most studied 2D examples, dichalcogenide 2H-TaSe2, and the formation of 
CDW is driven by a conventional nesting instability, which is masked by the pseudogap. Our 
findings reconcile and explain a number of unusual, as previously believed, experimental 
responses as well as disprove many alternative theoretical approaches12, 13, 14, 15. The 
magnitude, character and anisotropy of the 2D-CDW pseudogap are intriguingly similar to 
those seen in superconducting cuprates. 
Variations of the electron density in a metal are highly unfavourable because of the Coulomb 
repulsion. Though, some low-dimensional systems such as transition-metal dichalcogenides 
spontaneously develop a static periodic modulation (known as charge-density wave) of the electron 
gas below a certain temperature. A typical representative of the transition metal dichalcogenides is a 
2H (trigonal prismatic) polytype of TaSe2 which exhibits two CDW phase transitions at accessible 
temperatures: a second-order one at TNIC=122 K from normal to an incommensurate CDW state and 
a first-order  lock-in transition at TICC=90 K from the incommensurate to a 3x3 commensurate 
CDW phase16. In the one-dimensional case, where all points of the Fermi surface (FS) can be 
connected by the same vector (perfect nesting), the CDW transition occurs when the energy gain 
due to opening of a gap at the Fermi level exceeds the energy costs to distort the lattice2, 4, 17. In real 
2D materials the energy balance is more delicate since all FS points cannot be connected by the 
same vector (non-perfect nesting) and thus the FS can be gapped only partially. To understand the 
CDW mechanism in 2H-TaSe2, a detailed knowledge of the low-energy electronic structure is 
required. We therefore start with an overview of its temperature evolution in Figs. 1 and 2. 
The upper panel of Fig. 1 shows the topology of the Fermi surface in the normal state together with 
the CDW vectors Mqn Γ= 3
 defined by other experiments5, 16. Contrary to the earlier band 
structure calculations8, but in accordance with a recent study10, the FS consists of single hole-like 
“barrels” centred at Γ and K points, and electron-like “dogbones” around the M-point. The upper 
row of panels in Fig. 2a shows the corresponding dispersions of the electronic states crossing the 
Fermi level. FS sheets originate from two bands: one is responsible for the Γ and K barrels with a 
saddle point in between, the other one supports the dogbone with M being another saddle point. The 
second row of panels in Fig. 2a showing the data below the first phase transition, suggests that the 
normal ( 290 K ) and incommensurate CDW state ( TICC < 107 K < TNIC ) dispersions are 
qualitatively similar, except for the naturally different temperature broadening and weaker crossing 
#3 (Fig. 2a). This seems to be in agreement with the surprising earlier ARPES results, when 
virtually no change of the electronic structure has been detected upon entering the incommensurate 
CDW phase9, 10, 18, thus clearly contradicting the concept of the energy gain that should accompany 
a CDW transition. In contrast, the lock-in transition to the commensurate CDW state at TICC=90 K 
is much more pronounced (see lower panel of Fig.1). The new folded FS is schematically shown in 
Fig. 1 as a set of nearly circles around new Γ’-points and rounded triangles around new K’-points. 
This topology of the folded FS, though natural, as suggested by the normal state FS (see 
Supplementary material), has never been detected before. A possible reason could be a rather weak 
umklapp potential due to small lattice distortion of the order of 0.05 Å (Ref 16, 17), which 
consequently results in the intensity distribution along the FS that still reminds the one seen in the 
normal state.  Note, that without such an overview of the large portion of the k-space, it is 
problematic to understand what exactly happens to the electronic structure below 90 K. The energy-
momentum intensity distributions in the lower row of panels in Fig. 2a are also considerably 
modified by the 3x3 folding and show clear signatures of a strong hybridization. Exactly this 
hybridization, when occurs in the vicinity of the Fermi level, can lower the energy of the system 
(see right inset to Fig. 3c). This observation of strong changes at TICC is again paradoxical. 
According to other experimental techniques, the CDW phase transition with the critical energy 
lowering occurs at TNIC=122 K as is clearly seen in e.g. temperature dependences of the specific 
heat or resistivity6, 7. The lock-in transition at TICC=90 K, in contrast, is hardly detectable in the 
mentioned curves6, 7 and appears as a small break in the superlattice strength as seen by neutron 
scattering16.  
More detailed data analysis clarifies the situation. ARPES offers a unique opportunity to find out 
whether the energy gap opens up at the Fermi level anywhere on the FS by tracking the binding 
energy of the leading edge of an energy distribution curve (EDC) taken at kF. In Fig. 2b we show 
EDCs corresponding exactly to kF for selected high symmetry cuts in the k-space (Fig. 2a). Both 
panels unambiguously signal the leading edge shift (~15 meV) of the EDC #6. In both cases a clear 
suppression (not absence) of the spectral weight at the Fermi level is evident also from the 
corresponding energy-momentum distributions (rightmost panels in Fig. 2a), which is reinforced  by 
an obviously more diffused appearance of the K-barrels on the normal-state FS map (Fig. 1). In a 
striking analogy with the superconducting cuprates, we thus conclude the presence of a pseudogap 
in both, normal and incommensurate CDW states of  2H-TaSe2.  
In order to understand whether necessary energy gain can come from the pseudogap, we further 
characterize the pseudogap as a function of temperature and momentum in Fig. 3. The so called 
maps of gaps19, the plots of binding energies of the EDC leading edges as a function of k (i.e. not 
only kF),  are shown for the normal and incommensurate CDW states in Fig. 3a,b. These maps are a 
visual demonstration of the energy lowering of the system and are ideal for the determination of the 
anisotropy of the gap as they make the analysis of the behaviour of the leading edge in the vicinity 
of the Fermi surface (shown as dotted lines) possible. While the normal state map (Fig. 3a) reveals 
an isotropic pseudogap detectable only on the K-barrel, in the CDW state a pseudogap opens, in 
addition, on the dogbone FS and is anisotropic. As a quantitative measure of the pseudogap we take 
the difference of the leading edge binding energies of the kF-EDCs of M-dogbone and K-barrel 
which belong to the M-K cut, as is sketched in the left inset to the Fig. 3c. The sharp reproducible 
increase of the pseudogap magnitude below ~122 K, which escaped the detection before, is 
distinctly seen in Fig. 3c. It is now clear that it is the NIC transition (TNIC=122K) at which the 
critical lowering of the electronic energy occurs, and not only because of the larger pseudogap on 
the K-barrel, but also owing to the opening of the anisotropic pseudogap on the dogbone M-centred 
FS. In the commensurate CDW phase one can no longer characterize the energy gap by the leading 
edge position because of the interference with the folded bands --- the rounded corners of the new 
small triangular FS fall right where the normal state K-barrel was located (see Fig. 1). Instead, we 
plot in Fig. 3c also the values determined as shown in the right inset. The “band-gap” is a direct 
consequence of the hybridization, distinctly observed  below TICC (Figs. 1, 2). In order to emphasize 
the existence of a crossover regime where the pseudogap evolves into a band-gap we also plot in a 
limited T- interval the leading edge positions below TICC and the band-gap above TICC. Both gaps 
do not exhibit any anomalies at TICC monotonically increasing upon cooling deeper in the CDW 
state. 
Presented data already put certain limitations on several alternative theoretical approaches that were 
stimulated by the earlier experiments. The locations of the saddle points in the momentum-energy 
space obviously do not support the saddle point nesting scenario13 , proposed as an alternative to the 
conventional nesting, as both  points are far from the Fermi level (280 meV and 330 meV) and are 
not connected by any of the CDW vectors (the first one is located between Γ- and K-barrels, and the 
second is the M-point itself) . The pseudogap supported by the K-centred FS stays isotropic (within 
~ 2 meV) thus ruling out a six-fold symmetric CDW gap with nodes as suggested in Ref. 14. The 
value of the band-gap saturates at low temperatures at ~33 meV which is nearly a factor of 5 
smaller than the one (~ 150 meV) obtained in the strong-coupling approach12. A recent proposal15 to 
consider two components of the electronic structure, one of which is gapped and the other one is 
not, seems to be not supported by the data as well. 
What is then responsible for the CDW in two dimensions? In the following we demonstrate that the 
conventional FS nesting scenario, though modified by the presence of the normal-state pseudogap, 
is perfectly applicable.  From our high-resolution measurements, we have analysed the nesting 
properties of the FS quantitatively, which made it possible to explain the temperature evolution of 
the electronic structure of TaSe2 step by step. The quantitative measure of the nesting is the charge 
susceptibility. Here we approximate the charge susceptibility by the autocorrelation of the FS 
map20. In order to avoid the influence of the matrix elements, for further processing we take the 
model FS map shown in Fig. 4a, which is an exact copy of the experimental one as far as the locus 
of kF-points is concerned. In addition, this gives possibility to investigate the FS nesting properties 
of a hypothetical compound, with the electronic structure that yet have not reacted to the nesting 
instability (i.e. without gaps). The resulting autocorrelation maps as a function of temperature are 
shown in Fig. 4b, while the corresponding cuts along the ΓM direction -- in Fig. 4d. The sharp peak, 
seen in the 290 K curve exactly at 2/3 ΓM in Fig. 4d, is the first clear evidence for the nearly perfect 
nesting in 2D chalcogenides. This is at variance with the previous calculations17 which have found 
susceptibility for 2H polytypes to take a broadly humped form. Thus, the Fermi surface shape alone, 
without taking into account the pseudogap, results in the peak in the charge susceptibility at a wave 
vector ~2/3ΓM. According to the neutron scattering16, exactly at this wave vector there is a strong 
Kohn-like anomaly4 of the Σ1 phonon branch already at 300 K, and the matching phonon softens 
even more as the transition is approached. We consider therefore the formation of both, normal state 
pseudogap and anomaly of the Σ1 phonon branch as respective reactions of the electronic and lattice 
subsystems to the instability caused by the strong scattering channel that appears due to the nesting 
and a presence of the suitable mediating phonon. Such a mutual response can signify a strong 
electron-phonon interaction in 2H-TaSe2. It is interesting, that despite the favourable conditions, the 
system does not escape the instability by developing a static commensurate CDW order, 
presumably because of still too high temperature, which would effectively close not large enough 
band-gaps. Instead, it opens up a pseudogap. Upon cooling we observe correlated changes of the 
susceptibility, pseudogap magnitude, and energy of the softened phonon16: the peak in the 
susceptibility splits into two (middle panel in Fig. 4d), the pseudogap slowly closes (as suggested 
by the EDC shift at 290 K in Fig. 2b and a weak but detectable trend of the gap to decrease in Fig. 
3c) , the phonon energy becomes lower. This worsening of the nesting conditions is fundamentally 
different from the 1D case, where the susceptibility exhibits a peak, which becomes sharper with 
lowering the temperature4. We have found out that such anomalous behaviour of the susceptibility, 
i.e. the splitting and the size of the peak at ~2/3ΓM, is very sensitive to the shape of the Fermi 
surface itself, namely to the distance between the dogbone and K-centred FSs (distance, marked ‘D’ 
in Fig. 4a). The data plotted in Fig. 4c clearly suggest that the FS itself is temperature dependent 
explaining the variation of the charge susceptibility. Moreover, a close correlation with the 
temperature dependence of the pseudogap magnitude (Fig. 3c) implies that the pseudogap itself is 
directly related to the charge susceptibility. Nevertheless,  the transition finally occurs at TNIC=122 
K ,  but now only into the incommensurate CDW phase as dictated by the split peak in the 
susceptibility. Further reduction of the temperature results in a reversed modification of the 
obviously correlated quantities: pseudogap opens up more, peaks in susceptibility move towards 
each other (right panel in Fig. 4d), which agrees well with the dynamics of the superlattice peaks 
seen by neutrons, and phonon energy starts to increase again16. In such a manner the system arrives 
at the transition into the commensurate CDW state at 90 K. 
Our findings provide a natural explanation not only for the neutron scattering experiments16. Nearly 
linear in-plane resistivity in the normal state6, 7  is similar to the resistivity of an optimally doped 
cuprate superconductor21 and shows a typical behaviour of a pseudo-gapped metal. Below 122 K 
the slope is increasing in close correspondence to the larger pseudogap, and below 90 K the 
resistivity resembles the one of a normal metal. Optical measurements on the same single crystals7 
have  indirectly suggested the presence of a pseudogap already at 300 K as well, though with 
somewhat different energy scale. We also notice an excellent agreement with the Hall coefficient 
measurements22. According to Ref.22, the Hall coefficient starts to decrease sharply from its 
positive value below ~120 K, and changes sign at 90 K. The positive value in the normal state is 
explained by the larger volume of the Γ- and K-centred hole-like barrels in comparison with the 
electron-like dogbones around the M-points. The sharp increase of the pseudogap at 122 K results 
in the reduction of the number of charge carriers and finally in the commensurate phase (below 90 
K), the area enclosed by the electron-like circular FS around new Γ’-points is clearly larger than the 
two (per new BZ) hole-like triangular FS, which leads to the negative sign of the Hall coefficient. 
It was suggested before23 that the HTSCs and 2D chalcogenides have similar phase diagrams and 
that the pseudogap regime in cuprates is very similar to the CDW regime of chalcogenides. Now, 
after observation of the pseudogap in TaSe2 we can directly compare both pseudogaps. The energy-
momentum distribution of the photoemission intensity near kF of the K-barrel (crossing #6, Fig. 2a) 
is very similar to the one measured for the bonding barrel in the underdoped Bi2212-cuprate24, 25. In 
both cases one can still track the dispersion up to the Fermi level, but the spectral weight is 
significantly suppressed resulting in the shift of the kF-EDC’s leading edge. Furthermore, the 
detected anisotropy of the pseudogap on the dogbone FS is reminiscent of the famous anisotropic 
behaviour of the pseudogap in cuprates. Finally, the highly inhomogeneous intensity distribution 
along some of the small triangular FSs and especially along incomplete circles around new Γ’ 
points cannot escape a comparison with the famous Fermi surface ‘arcs’. We believe that this series 
of remarkable similarities to the high-temperature superconducting cuprates calls for more careful 
comparative studies of the pseudogap phenomenon in these materials. 
It is interesting, that unlike in the 1D case where a pseudogap has been reported26, 27, 28 to be a 
consequence of fluctuations (potentially able to suppress the transition temperature up to a quarter 
of the mean-field value29), the pseudogap in 2D shows unusual non-monotonic behaviour clearly 
tracking the temperature evolution of both, the bare susceptibility and phonon spectrum, and thus 
seems to represent a natural response of the system to a nesting instability.  
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Acknowledgements: The project was supported, in part, by the DFG under Grant No. KN393/4. We thank R. Hübel for 
technical support. This work was partially performed at the Swiss Light Source, Paul Scherrer Institut, Villigen, 
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Correspondence and requests for materials should be addressed to S.V.B. (S. Borisenko@ifw-dresden.de). 
Figure 1. Fermi surfaces. Momentum distribution of the photoemission intensity at the 
Fermi level at 180 K and 30 K. Short dashed lines are the BZ boundaries. 
Figure 2. Electronic structure and leading edge gap. a) Photoemission intensity as a 
function of energy and momentum in the normal (upper row of panels), incommensurate 
CDW (middle row) and commensurate CDW (lower row) states. Sketches of the FS on the 
middle panels show cuts in momentum space along which the data were taken and are 
valid for all panels in the same column. Hybridization effects in the lower row are seen as 
“repulsions” of the bands, which occur when a folded band is supposed to cross the 
original one, as schematically shown in the right inset to Fig. 3b. Since the spectral weight 
of the folded band is lower, these effects appear as breaks in the intensity of the original 
bands. Numbers correspond to the different kF. b) kF EDCs from the datasets shown in a). 
Leading edge gap is clearly seen as a shift of the EDC#6 to the higher binding energies at 
290 K and 107 K. 
Figure 3. Momentum and temperature dependence of the pseudogap. a,b) Binding 
energies of the leading edges of all EDCs from the momentum range close to the 
irreducible parts of the M-dogbone and K-barrel FS. Colour scales reproduce the vertical 
coordinate. Anisotropy of the pseudogap on the M-dogbone FS is seen as a changing 
colour when going along the dashed line which correspond to FS, i.e. kF points. c) 
Difference between the binding energies of the leading edges of the EDC#5 and #6 from 
Fig. 2b as a function of temperature as shown schematically in the left inset (pseudogap) 
when cycling the temperature (filled symbols). Shift of the EDC maximum (bandgap) which 
corresponds to the top of the hybridized band as shown in the right inset with respect to its 
position at TNIC=122 K (open symbols). Note that the given leading edge gap values below 
90 K cannot be considered as a measure of the pseudogap because of folding. 
Figure 4. Nesting properties. a) A model copy of the FS map from Fig. 1 with the 
homogeneous intensity distribution along the FS. b) Autocorrelation maps of this model at 
different temperatures. c) Temperature dependence of the momentum distance between 
the M-dogbone and K-barrel (T > 90 K). The distance below 90 K is just a difference 
between corresponding maxima at the FS map. d) Intensity of the maps from b) along the 
cuts shown by dotted lines. 
Supplementary information. 
Here we explain in more details the nesting properties of the normal state FS and how to understand 
the 3x3 folding upon entering the commensurate CDW regime. 
In panel a we show only Γ- and K-centred barrels and their first order ( 1q
± , 2q
± , 3q
± ) and 
second order ( 21 qq
+ , 32 qq
+ )  replica. One can see that the nesting conditions are nearly perfect 
for all points of the K-barrel: shifting the Γ-barrel by one of the CDW vectors results in an overlap 
with the corresponding part of the K-barrel with opposite Fermi velocities. Panel b shows an 
idealized situation, which is actually very close to the real one, since a more careful examination of 
Fig. 1a reveals the straight sections of both Γ- and K-barrels.  
This is, however, not enough to explain the fully gapped K-barrel in the CDW state.  
In panel a it is seen that 3x3 folding produces a double-walled barrels in the centre of the new BZ 
which are supposed to interact with the single copies of Γ-barrel. This is not surprising as the 
original BZ contains only one complete Γ-barrel and two complete K-barrels. It means, that the K-
barrels of a system with the FS schematically shown in panel b, will not disappear after a 3x3 
folding. 
Panel c shows the dogbone FS centred around M-points together with its first- and second order 
replica. Folding of this FS sheet results in the same triangular FSs as in panel a, a rather 
complicated set of features around the centres of the new BZ and nearly exact copies of the Γ-
barrels and its replica. It is these copies of the Γ-barrels which were missing in panel a to interact 
with double-walled K-barrels.  
Panel d, where all FSs and their folded replica are shown, summarizes all mentioned above. When 
hybridization effects are switched on, the former K-barrel completely disappears, doubly degenerate 
triangular FSs emerge around the corners of the new BZ and the complicated set of features seems 
to evolve into a four-times degenerate nearly circle FSs around the Γ’- points (at least 
experimentally we can currently resolve only these semi-circular FSs and not two doubly 
degenerate hexagons with rounded corners as is suggested by the panel c).
