Coexistence of Interacting Charge Density Waves in a Layered Semiconductor

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 ${\mathrm{EuTe}}_4$, a layered crystal that has drawn considerable attention owing to its anomalous thermal hysteresis and a semiconducting CDW state despite the absence of perfect Fermi surface nesting. By accessing unoccupied conduction bands with time- and angle-resolved photoemission measurements, we find that monolayers and bilayers 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 ${\mathrm{EuTe}}_4$ but also provide a general nonequilibrium approach to understand coexisting, layer-dependent orders in a complex system.

Figure 1

Comparison between charge density waves in $R{\mathrm{Te}}_3$ and ${\mathrm{EuTe}}_4$. (a) Crystal structure of $R{\mathrm{Te}}_3$, featuring Te bilayers whose states are closest to the Fermi level. (b) Calculated normal-state FS (blue lines) of $R{\mathrm{Te}}_3$ 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 the folded bands are plotted for clarity. (c) Temperature-dependent electrical resistivity of $R{\mathrm{Te}}_3$, showing metallic behavior in the CDW state. The CDW transition temperatures of all the compounds shown are above 300 K. (d) Crystal structure of ${\mathrm{EuTe}}_4$. (e) Tight-binding normal-state FS of ${\mathrm{EuTe}}_4$, following the same convention as in panel (b). (f) Temperature-dependent electrical resistivity of ${\mathrm{EuTe}}_4$ in the CDW state, showing semiconducting behavior and a giant thermal hysteresis. Panels (c) and (f) are extracted from Refs. [12] and [13], respectively.

Figure 2

Electronic structure and intrinsic CDW gaps of ${\mathrm{EuTe}}_4$. (a) Original (blue curves) and folded (red curves) Fermi surface based on tight-binding calculations. 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 $t=0\mathrm{ps}$. 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 $0.16\mathrm{mJ}/{\mathrm{cm}}^2$.

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.

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 $1.2\mathrm{mJ}/{\mathrm{cm}}^2$. (c) The calculated band structure along the $\mathrm{\Gamma }\text{−}X$ 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.