††thanks: These authors contributed equally to this work: B. Q. Lv and Alfred
Zong.††thanks: These authors contributed equally to this work: B. Q. Lv and
Alfred Zong.

# Coexistence of interacting charge density waves in a layered semiconductor

B. Q. Lv  Tsung-Dao Lee Institute, Shanghai Jiao Tong University, Shanghai
200240, China.  Massachusetts Institute of Technology, Department of Physics,
Cambridge, MA 02139, USA.  School of Physics and Astronomy, Shanghai Jiao Tong
University, Shanghai 200240, China.  Zhangjiang Institute for Advanced Study,
Shanghai Jiao Tong University, Shanghai 200240, China.  Alfred Zong
Massachusetts Institute of Technology, Department of Physics, Cambridge, MA
02139, USA.  University of California at Berkeley, Department of Chemistry,
Berkeley, CA 94720, USA.  Dong Wu  Beijing Academy of Quantum Information
Sciences, Beijing 100913, China.  Zhengwei Nie  Beijing National Laboratory
for Condensed Matter Physics and Institute of Physics, Chinese Academy of
Sciences, Beijing 100190, China.  Yifan Su  Massachusetts Institute of
Technology, Department of Physics, Cambridge, MA 02139, USA.  Dongsung Choi
Massachusetts Institute of Technology, Department of Electrical Engineering
and Computer Science, Cambridge, MA 02139, USA.  Batyr Ilyas  Massachusetts
Institute of Technology, Department of Physics, Cambridge, MA 02139, USA.
Bryan T. Fichera  Massachusetts Institute of Technology, Department of
Physics, Cambridge, MA 02139, USA.  Jiarui Li  SLAC National Accelerator
Laboratory, Menlo Park, CA 94025, USA.  Department of Applied Physics,
Stanford University, Stanford, CA 94305, USA.  Edoardo Baldini  Massachusetts
Institute of Technology, Department of Physics, Cambridge, MA 02139, USA.
Masataka Mogi  Massachusetts Institute of Technology, Department of Physics,
Cambridge, MA 02139, USA.  Y.-B. Huang  Shanghai Synchrotron Radiation
Facility, Shanghai Advanced Research Institute, Chinese Academy of Sciences,
Shanghai 201204, China.  Hoi Chun Po  Department of Physics, Hong Kong
University of Science and Technology, Clear Water Bay, Hong Kong, China.
Sheng Meng  Beijing National Laboratory for Condensed Matter Physics and
Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China.  Yao
Wang  Department of Chemistry, Emory University, Atlanta, GA 30322, USA.  N.
L. Wang  Beijing Academy of Quantum Information Sciences, Beijing 100913,
China.  International Center for Quantum Materials, School of Physics, Peking
University, Beijing 100871, China.  Nuh Gedik<EMAIL_ADDRESS>Massachusetts
Institute of Technology, Department of Physics, Cambridge, MA 02139, USA.

###### Abstract

Coexisting orders are key features of strongly correlated materials and
underlie many intriguing phenomena from unconventional superconductivity to
topological orders. Here, we report the coexistence of two interacting charge-
density-wave (CDW) orders in EuTe4, a layered crystal that has drawn
considerable attention owing to its anomalous thermal hysteresis and a
semiconducting CDW state despite the absence of perfect FS nesting. By
accessing unoccupied conduction bands with time- and angle-resolved
photoemission measurements, we find that mono- and bi-layers of Te in the unit
cell host different CDWs that are associated with distinct energy gaps. The
two gaps display dichotomous evolutions following photoexcitation, where the
larger bilayer CDW gap exhibits less renormalization and faster recovery.
Surprisingly, the CDW in the Te monolayer displays an additional momentum-
dependent gap renormalization that cannot be captured by density-functional
theory calculations. This phenomenon is attributed to interlayer interactions
between the two CDW orders, which account for the semiconducting nature of the
equilibrium state. Our findings not only offer microscopic insights into the
correlated ground state of EuTe4 but also provide a general non-equilibrium
approach to understand coexisting, layer-dependent orders in a complex system.

Quantum materials driven by nonperturbative correlations display rich phase
diagrams thanks to multiple instabilities at similar energy and time scales
[1]. The microscopic understanding of macroscopic manifestations of the
resulting coexistent orders has been a central subject in condensed matter
physics. In solids, order formation due to symmetry-breaking is often
accompanied by the appearance of an energy gap near the Fermi level, where the
gap size is proportional to the amplitude of the order parameter. A prominent
example is the charge-density-wave (CDW) order. Historically, Fermi surface
(FS) nesting — the matching of sections of the FS to some other parts by a
single wave vector $\mathbf{q}$ — was suggested as the main driver for CDW
formation in one-dimensional chains [2, 3], leading to a phase transition from
the metallic to semiconducting state. For quasi-two-dimensional materials,
however, a perfect nesting condition can seldom be fulfilled due to the
complex FS topology, and other mechanisms are often at play for the CDW
formation [4, 5]. Hence, in quasi-2D materials, the CDW energy gap usually
does not open everywhere on the FS and the system remains metallic even in the
CDW state. One well-established example is the family of $R$Te3 ($R=$
lanthanide except Pm, Eu, Yb, and Lu) [6, 7, 8, 9], which is known for the CDW
states originating from the conducting Te bilayers [Fig. 1(a)]. The normal-
state FS can be well approximated by a tight-binding model that considers the
Te bilayers [10], as shown in Fig. 1(b). One can see that the FS favors a good
but not perfect nesting, leading to the metallic CDW state with a partially
gapped FS, as evidenced by metallic behavior in electrical transport [Fig.
1(c)] and remnant FS seen in angle-resolved photoemission spectroscopy (ARPES)
[11, 12, 10].

Figure 1: Comparison between charge density waves in RTe${}_{\text{3}}$ and
EuTe${}_{\text{4}}$. (a) Crystal structure of $R$Te3, featuring Te bilayers
whose states are closest to the Fermi level. (b) Calculated normal-state FS
(blue lines) of $R$Te3 based on the tight-binding model. The blue solid and
red dashed lines represent the original and folded bands, respectively. Note
that only parts of folded bands are plotted for clarity. (c) Temperature-
dependent electrical resistivity of $R$Te3, showing metallic behavior in the
CDW state. The CDW transition temperatures of all the compounds shown are
above 300 K. (d) Crystal structure of EuTe4. (e) Tight-binding normal-state FS
of EuTe4, following the same convention as in panel (b). (f) Temperature-
dependent electrical resistivity of EuTe4 in the CDW state, showing
semiconducting behavior and a giant thermal hysteresis. Panels (c) and (f) are
extracted from ref. [12] and ref. [13], respectively.

The recently synthesized Eu-based telluride EuTe4, where Eu is divalent
instead of the trivalent lanthanide in $R$Te3, has stimulated intense interest
due to its unique electrical transport property: an anomalously large thermal
hysteresis embedded in a semiconducting CDW state [14, 13, 15, 16, 17, 18,
19], shown in Fig. 1(f). EuTe4 shares key structural motifs with $R$Te3, both
consisting of an alternate stacking of the insulating $R$Te spacers and the
nearly-square Te nets [Fig. 1(a),(d)]. As expected, the calculated normal-
state FS of EuTe4 exhibits imperfect nesting, as illustrated in Fig. 1(e).
These similarities point to a metallic CDW state, which is also suggested by
density-functional theory (DFT) calculations [14, 15]. However, previous ARPES
studies revealed a fully gapped FS [13, 16, 17]. The stark contrast raises the
puzzle of why the CDW state of EuTe4 is semiconducting. Structurally, the
apparent difference is that EuTe4 contains both Te monolayers and bilayers in
a single unit cell, hinting at the possibility of coexistence of layer-
dependent CDW orders. Therefore, one important clue is the additional
interactions between the two CDW orders. Yet, despite deploying multiple
experimental probes [13, 16], it remains unclear whether more than one CDW
order exists in EuTe4 in the first place. In this work, we leverage the
capability of time-resolved ARPES (tr-ARPES) [20, 21, 22, 23] to distinguish
and probe the optical excitation of two CDW orders in EuTe4. Our findings
suggest that the interlayer CDW coupling underpins the semiconducting nature
of EuTe4, and our methodology offers a promising route for visualization and
ultrafast manipulation of coexisting orders in other quasi-2D systems.

Figure 2: Electronic structure and intrinsic CDW gaps of EuTe${}_{\text{4}}$.
(a) Tight-binding original (blue lines) and folded (red lines) Fermi surface.
The solid lines represent the observed bands at $E_{v}$ [shown in panel (b)].
The black dashed circle indicates the detection area of our tr-ARPES
measurements. (b) ARPES constant energy map at $E_{v}$, measured with 90 eV,
right circularly polarized light. (c) ARPES (i) and curvature (ii) intensity
plots along the C1 direction [labeled in panel(b)], measured at $t=0$ ps. The
probe photon energy was 10.75 eV, and polarization was linear horizontal (LH,
perpendicular to the photoemission plane). The pump photon energy was 1.55 eV,
the polarization was LH, and the fluence was 0.16 mJ/cm2.

We first examine the occupied valence bands of EuTe4 in its CDW state. The
calculated normal-state FS, as shown by the blue curves in Fig. 1(e), is
mainly composed of two perpendicular sets of quasi-parallel curves. In the CDW
state, these curves are folded along the $k_{y}$ direction, leading to
additional Fermi pockets in the Brillouin zone [see red curves in Fig. 2(a)].
We measured the fermiology in the CDW state by high-resolution synchrotron-
based ARPES at 20 K. Figure 2(b) displays a representative constant-energy map
at $E_{v}$ [indicated in Fig. 2(c)], where both the original bands and the
folded CDW bands are resolved. The measurement in Fig. 2(b) is largely
consistent with the tight-binding prediction in Fig. 2(a) except for the
missing pockets at $|k_{x}|\lesssim 0.5\pi/a$ [indicated by dashed curves in
Fig. 2(a)], which are the consequence of large energy gaps in the CDW state,
as we will quantify later.

Figure 3: Temporal evolution of conduction bands and CDW gaps. (a)–(e) ARPES
(i) and curvature (ii) intensity plots along the C1 direction at $t=0$, 0.2,
0.4, 0.6, and 0.9 ps, respectively. The red symbols are peak positions of Con2
and Con3 bands, obtained from the Lorentzian fitting of the energy
distribution curves at each $k_{x}$point and at each delay time. (f) Extracted
time-dependent conduction band bottoms. The inset shows the corresponding CDW
gaps determined from the difference between the bottom of Con2 or Con3 and the
top of V0. (g) Time-dependent band-bottom difference of Con1 and Con2. The
error bars represent the statistical uncertainty derived from the fitting
procedure.

To discern the possible existence of two CDWs, information about unoccupied
single-particle states must be characterized, which gives a more quantitative
measure of the CDW energy gap and hence reflects the intrinsic order parameter
amplitude. These unoccupied states are difficult to detect in equilibrium
ARPES due to the large gap size compared to thermal energy at experimentally
accessible temperatures up to $\sim 400$ K [13]; inverse ARPES would also be
unsuitable due to the coarse energy resolution. The limitations of the
equilibrium techniques necessitate the use of tr-ARPES to differentiate CDW
orders in EuTe4. With this in mind, we conducted tr-ARPES measurements using
an above-gap infrared (1.55 eV) pump laser and an extreme-ultraviolet (XUV,
10.75 eV) probe laser (see [24] for experimental details). Figure 2(c) shows
the measured transient electronic structure along $\Gamma$–X under an incident
fluence of 0.16 mJ/cm2 at $t\sim 0$ ps (i.e., at pump-probe temporal overlap).
In this weak perturbation regime such that Floquet states can be neglected,
pump pulses are expected to minimally disturb the CDW structure at $t\sim 0$
ps. As a result, the intensity that indicates populated states above $E_{F}$
more closely resembles the unoccupied states in equilibrium compared to
spectra at later time delays [25], giving us access to the unoccupied band
structure that is otherwise inaccessible in equilibrium.

Three spectral features, labeled as Con1, Con2, and Con3 in Fig. 2(c), can be
identified above $E_{F}$, indicating three conduction-band bottoms in the CDW
state. Con1 resides at the $\Gamma$ point while Con2 and Con3 are close to
each other in crystal momentum along the $\Gamma$–X direction but are
energetically distinct. To trace the origins of these unoccupied bands, we
compare Fig. 2(c) with Fig. 2(a) and the calculated normal-state electronic
structure in Fig. S6(a). It is evident that Con1 can only be assigned to the
folded CDW band since no normal-state bands lie within an energy window of
$\pm 1$ eV at $\Gamma$. By contrast, Con2 and Con3 are parts of the normal-
state main bands with a gap opening near $E_{F}$. By calculating the
difference between the energy minima of Con2 and Con3 and the energy maximum
of the valence band top [labeled as V0 in Fig. 2(c)], we estimate the
corresponding CDW gaps along $\Gamma$–X to be 0.8 eV and 1.0 eV, respectively.
The presence of two distinct single-particle gaps hence demonstrates the
coexistence of two CDW orders.

Spectroscopically, separated bands like Con2 and Con3 do not always guarantee
two distinct orders because scenarios such as spin-orbit coupling can also
lead to band splitting. To rule out other scenarios and to pin down the
coexistence of the two CDW orders, we study the evolution of their spectral
functions after photoexcitation. Figures 3(a)–(e) show the snapshots of
photoemission spectra above $E_{F}$ at five representative pump-probe delay
points (first row) and their respective curvature plots that highlight the
band positions (second row). Besides the time-dependent spectral weights due
to photo-induced carrier excitation and recombination, the energy positions of
all three conduction bands Con1–Con3 shift down immediately after the pump
pulse arrival, indicating an overall non-equilibrium suppression of the CDW
order. Importantly, there is a larger energy separation between Con2 and Con3
in Fig. 3(e) compared to Fig. 3(a) (highlighted by the dashed arrows),
suggesting two different evolution pathways for the two bands. To quantify
this distinction, we fit the energy distribution curves (EDCs) between $k_{1}$
and $k_{2}$ labeled in Fig. 3(a)i with Lorentzian functions (see [24]). The
extracted momentum-dependent EDC peak positions of Con2 and Con3 are
superimposed on the curvature plots [Fig. 3(a)ii–3(e)ii] with red markers,
which very well match the band intensity and hence affirm the validity of both
the EDC fits and the curvature plots. We summarize the fitting results in Fig.
3(f). Con2 and Con3 clearly exhibit different dynamics. The bottom of Con3
decreases from 0.78 eV at $t=0$ to 0.71 eV at 0.4 ps, and then starts to
recover after 0.4 ps. By contrast, the bottom of Con2 keeps decreasing until
0.9 ps. After 0.9 ps, both Con2 and Con3 are undetectable [see Fig. S4] due to
fast excited carrier relaxation and hence the disappearance of spectral
weights. These distinct dynamics lend further proof that Con2 and Con3 come
from two different orders.

We can associate the above observations of two CDW gap sizes and two
qualitatively different photoinduced dynamics with the unique CDW
instabilities and structural motifs in EuTe4, which feature both Te monolayers
and Te bilayers. Due to the quasi-2D nature of EuTe4, strong hybridization of
states between the mono- and bi-layers are limited, so it is reasonable to
attribute the two CDW orders to the monolayer and bilayer Te sheets. To obtain
the specific assignment of Con2 and Con3, we note that within a Te bilayer,
the additional intra-bilayer interaction can further enhance the broken
symmetry order, as has been verified by our DFT results in Fig. S6. Hence, the
CDW gap associated with Te bilayers is expected to be larger, and we assign
Con3 and Con2 to the bilayer and monolayer CDW, respectively. This assignment
was further supported by the observed dichotomous temporal evolutions in Fig.
3. The stronger bilayer CDW experiences less suppression and hence a faster
recovery (Con3), whereas the weaker monolayer CDW has a larger gap suppression
and delayed recovery [Fig. 3(f)]. These characteristics not only provide
strong evidence for the existence of monolayer and bilayer CDWs but also
suggest that photoexcitation can be an effective means for charting separate
pathways of their dynamics due to the different CDW strengths.

We next discuss the behavior of Con1, which, unlike Con2 and Con3, is a part
of the folded bands that are only present in the CDW state. We assign Con1 to
the CDW residing in the monolayer Te sheet for two reasons. First, Con1 is a
global conduction band minimum as directly measured in Fig. 3(a) and as
evidenced by the long population lifetime in Fig. S4, which results from the
slow equilibration of excited electrons located at the conduction band bottom
of a semiconductor [26]. Second, Con1 displays a similar dynamical evolution
as Con2, namely, both keep shifting downward toward larger binding energy over
$\sim 1$ ps. The main difference between Con1 and Con2 is the magnitude of the
shift, where Con1 exhibits a smaller downshift on average compared to Con2. As
a result, the difference between the energy bottoms of Con1 and Con2 decreases
as a function of time, as shown in Fig. 3(g). Such discrepancy is more
prominent under stronger laser excitation. In Fig. 4(a) and (b), we present
the transient electronic structure at $t=0.05$ and 0.75 ps after excitation
with 1.2 mJ/cm2, 800-nm pump pulses. Under such a strong perturbation, the
energy gap between Con2 and V0 almost completely collapses due to the large
downshift of Con2. By contrast, Con1 shows a much smaller downshift and it is
clear that the energy gap does not vanish at the $\Gamma$ point.

The observation of disproportionate gap renormalization of Con1 and Con2 is
unexpected because it conflicts with the previous understanding of EuTe4 that
there is only one order parameter responsible for gap opening. In other words,
one should expect a proportional downshift of Con1 and Con2 if the monolayer
CDW has a single CDW order parameter. To understand this behavior, we
performed detailed DFT calculations based on the $1\times 3\times 2$
superlattice, which is a good approximation of the in-plane incommensurate and
out-of-plane commensurate CDW structure [13, 14]. The DFT results, in good
agreement with previous reports [14, 27], are summarized in Fig. 4(c) and
[24]. One can see that the Te monolayer indeed has a relatively weaker CDW,
resulting in several conduction band bottoms, supporting our assignment of
Con1, Con2, and Con3 in Fig. 3. On the other hand, besides an overall shift of
the chemical potential, the major discrepancy between the DFT results and the
measured bands in Fig. 2(c) is that DFT yields a metallic FS, suggesting that
additional electron correlations beyond those captured by DFT must be
considered to explain the observed semiconducting CDW, especially for the
monolayer CDW gap at Con2.

Figure 4: The interplay of intertwined CDW states. (a),(b) ARPES (i) and
curvature (ii) intensity plots along the $k_{x}$ direction at $t=0.05$ and
0.75 ps, respectively. The pump fluence was 1.2 mJ/cm2. (c) The calculated
band structure along the $\Gamma$–X direction based on the ($1\times 3\times
2$) superlattice and the GGA type of the exchange-correlation potential. The
red and blue colors represent the weight of $p_{x}$ and $p_{y}$ orbitals from
the Te monolayer and bilayer, respectively. (d) Schematic of the electron-hole
interactions between CDWs residing in the monolayer and bilayer Te sheets. The
white circles and black dots represent the holes and electrons, respectively.
The orange curves denote the interactions, and the black-dotted ellipse
represents an electron-hole pair.

Motivated by the recent investigations of the insulating state in
1$T$-TaS${}_{\text{2}}$ induced by interlayer coupling [28, 29, 30, 31, 32],
we speculate that the experimentally observed gap in EuTe4 arises from mono-
and bi-layer CDW couplings. A direct observation from the DFT-calculated band
structure is that the low-energy conduction and valence bands are primarily
contributed by monolayer and bilayer Te orbitals, respectively (Fig. S6). The
energy proximity and spatial separation of electrons and holes make the
material susceptible to charge transfer between these nominally charge-neutral
Te layers [33]. Since DFT is a ground-state theory and the functionals do not
include excitonic effects, we can compensate for this missing interlayer
interaction by an additional term in the Hamiltonian,
$Vn^{(\alpha)}n^{(\beta)}$, where $n^{(\alpha)}$ and $n^{(\beta)}$ correspond
to the electron density operators at the monolayer and bilayer Te bands, and
$V$ represents the attractive interband coupling. A direct consequence is the
separation of these types of CDW bands [red and blue curves in Fig. 4(c)]
caused by the Hartree part of this interaction. Furthermore, this interaction
favors the formation of inter-band excitons, whose condensation opens an
excitonic gap. Both gapping mechanisms are directly driven by, and therefore
proportional to, $V$, without requiring a structural transition. It is
predicted that this effective interaction can be screened by carriers in both
layers, leading to a reduction of the aforementioned gap. This is consistent
with the experimentally observed shift of Con2 [see Fig. 4(d)] with the
presence of light-induced photocarriers. At the same time, the intralayer CDW
orders remain unchanged, manifesting as the persistence of the Con1,
highlighting the distinct origins of the CDW gap and the interband
interaction-driven gap.

Our time-domain spectroscopic investigations of the CDW state in EuTe4 unveil
the following key results. (i) In a single bulk crystal, the Te monolayer and
bilayer host distinct orders, with the bilayer experiencing a stronger CDW
distortion due to the additional intra-bilayer interactions. (ii) As a result,
the bilayer CDW is less renormalized and recovers faster under small
perturbations by light. (iii) The monolayer Con1 and Con2 conduction bands
exhibit disproportionate temporal evolutions, deviating from the typical
dynamics of a single-order parameter. We interpret this deviation and the
related semiconducting nature of the CDW as a result of additional mono-
bilayer interactions. These findings provide crucial information on the
dynamics and interplay of coexisting CDW orders, highlighting the importance
of interlayer coupling in semiconducting quasi-2D CDW systems. As a newly
discovered CDW material, EuTe4 also offers a rich platform for understanding
and manipulating the layer-specific CDW orders with a variety of external
parameters such as temperature and ultrashort light pulses, which can result
in anomalous electrical transport and persistent hidden states [13, 19]. The
above knowledge of coexisting CDW orders can also be generalized to other
layered systems, such as multilayer cuprates [34], and our study thereby
offers another route for elucidating the novel physics of coexisting orders in
these quantum many-body systems.

###### Acknowledgements.

We thank B. V. Fine, A. V. Rozhkov, J.-Z. Zhao, and D. Azoury for fruitful
discussions. We acknowledge support from the U.S. Department of Energy, Office
of Science, Office of Basic Energy Sciences, DMSE (instrumentation and data
taking), the National Science Foundation under Grant No. NSF DMR-1809815 (data
analysis), and the Gordon and Betty Moore Foundation’s EPiQS Initiative grant
GBMF9459 (manuscript writing). B.Q.L. acknowledges from the TDLI starting up
grant, the National Natural Science Foundation of China (23Z990202580), the
Ministry of Science and Technology of China (2023YFA1407400), the Shanghai
Natural Science Fund for Original Exploration Program (23ZR1479900), and the
Shanghai Talent Program. A.Z. acknowledges support from the Miller Institute
for Basic Research in Science. D.W. and N.L.W. acknowledge support from the
National Natural Science Foundation of China (No. 11888101), and the National
Key Research and Development Program of China (No. 2022YFA1403901). Y.B.H.
acknowledges support from the National Key Research and Development Program of
China (2017YFA0403401) and the National Natural Science Foundation of China
(U1875192, U1832202). Y.W. acknowledges support from U.S. Department of
Energy, Office of Science, Basic Energy Sciences, under Early Career Award No.
DE-SC0024524. E.B. acknowledges additional support from the Swiss NSF under
fellowships P2ELP2-172290 and P400P2-183842.

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