⇦Chinese Physics Letters, 2023, Vol. 40, No. 10, Article code 107402 Pressure-Induced Superconductivity in the Charge-Density-Wave Compound LaTe$_{2- x}$Sb$_{x}$ ($x = 0$ and 0.4) Xu Chen (陈旭)1†, Pei-han Sun (孙培函)2†, Zhenkai Xie (谢圳楷)1,3†, Fanqi Meng (孟繁琦)1, Cuiying Pei (裴翠颖)4, Yanpeng Qi (齐彦鹏)4,5,6, Tianping Ying (应天平)1, Kai Liu (刘凯)2*, Jian-gang Guo (郭建刚)1*, and Xiaolong Chen (陈小龙)1,3* Affiliations 1Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China 2Department of Physics and Beijing Key Laboratory of Opto-electronic Functional Materials & Micro-nano Devices, Renmin University of China, Beijing 100872, China 3University of Chinese Academy of Sciences, Beijing 100049, China 4School of Physical Science and Technology, ShanghaiTech University, Shanghai 201210, China 5ShanghaiTech Laboratory for Topological Physics, ShanghaiTech University, Shanghai 201210, China 6Shanghai Key Laboratory of High-resolution Electron Microscopy and ShanghaiTech Laboratory for Topological Physics, ShanghaiTech University, Shanghai 201210, China Received 14 July 2023; accepted manuscript online 5 September 2023; published online 11 October 2023 †These authors contributed equally to this work.
*Corresponding authors. Email: kliu@ruc.edu.cn; jgguo@iphy.ac.cn; chenx29@iphy.ac.cn Citation Text: Chen X, Sun P H, Xie Z K et al. 2023 Chin. Phys. Lett. 40 107402    Abstract Magnetic CeTe$_{2}$ achieving superconductivity under external pressure has received considerable attention. The intermingling of 4$f$ and 5$d$ electrons from Ce raised the speculation of an unconventional pairing mechanism arising from magnetic fluctuations. Here, we address this speculation using a nonmagnetic 4$f$-electron-free LaTe$_{2}$ as an example. No structural phase transition can be observed up to 35 GPa in the in situ synchrotron diffraction patterns. Subsequent high-pressure electrical measurements show that LaTe$_{2}$ exhibits superconductivity at 20 Gpa with its $T_{\rm c}$ (4.5 K) being two times higher than its Ce-counterpart. Detailed theoretical calculations reveal that charge transfer from the 4$p$ orbitals of the planar square Te–Te network to the 5$d$ orbitals of La is responsible for the emergence of superconductivity in LaTe$_{2}$, as confirmed by Hall experiments. Furthermore, we study the modulation of $q_{\scriptscriptstyle{\rm CDW}}$ by Sb substitution and find a record high $T_{\rm c}^{\rm onset} \sim 6.5$ K in LaTe$_{1.6}$Sb$_{0.4}$. Our work provides an informative clue to comprehend the role of $5d$–$4p$ hybridization in the relationship between charge density wave (CDW) and superconductivity in these RETe$_{2}$ (RE = rare-earth elements) compounds.
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 DOI:10.1088/0256-307X/40/10/107402 © 2023 Chinese Physics Society Article Text The coexistence of charge density wave (CDW) ordering, magnetism, and superconductivity discovered in CeTe$_{2}$ with successively lowering temperature has raised intensive research interests, which is reminiscent of the competitive nature between superconductivity and magnetism in many $f$-electron compounds such as CeCu$_{2}$Si$_{2}$, CeIn$_{3}$, and CeRhIn$_{5}$.[1-3] The ambient crystal structure of CeTe$_{2}$ is composed of a Te(1) square lattice, stacking with CeTe(2) layer alternatively along the $c$ axis,[4,5] as shown in Fig. 1(a). While the square Te(1) layer is suggested to be responsible for the emergence of superconductivity under high pressure,[6] the CeTe(2) layer experience a short-range ferromagnetic ordering at 6 K and a subsequent A-type antiferromagnetic ordering at 4.3 K.[6,7] From the charge-transfer point of view, CeTe$_{2}$ can be written as [Ce$^{3+}$Te(2)$^{2-}$]Te(1)$^{1-}$, with an empty 5$d$ orbital and one leftover 4$f$ electron of Ce that is responsible for the emergence of rich magnetic ordering.[8] The coexistence of long-range magnetic ordering with superconductivity raises the speculation of a magnetism-mediated electron pairing in the present system and a spin-triplet ground state.[6] This is further complicated by the accompanied CDW ordering.[9] However, the scenario of magnetic pairing is challenged by the relatively small upper critical magnetic field of 0.6 T, far below the Pauli limit.[6] An angle-resolved photoemission spectroscopy (ARPES) investigation discovered a negligible contribution of Ce 4$f$ electrons near the Fermi level ($E_{\rm F}$).[8] Later, detailed theoretical analyses show that the charge transfer from Te(1) 5$p$ orbitals to the Ce 5$d$ states could effectively enhance the density state at $E_{\rm F}$ and may serve as the pairing glue of the electrons.[8] Further calculations suggest that electron doping raising from Te vacancies could fill the CDW gap, and the evolution of the Fermi surface (FS) cannot be described under a simple rigid band scheme.[10-12] On the other hand, the 4$f$-electron-free LaTe$_{2}$ is predicted to have almost identical band structures and a comparable $T_{\rm c}$.[8] However, all the above theoretical calculations are based on the assumption of isostructural compression with no phase transition under external pressure. The key information of structural evolution under high pressure is missing up to date. It is therefore imperative to measure the high-pressure transport behavior of LaTe$_{2}$, as well as its crystal structure, without any interference from the magnetic ordering of 4$f$ orbitals. In this Letter, we carry out a systematic characterization of single crystalline LaTe$_{2}$ and LaTe$_{1.6}$Sb$_{0.4}$. In situ synchrotron diffractions substantiate that the initial crystal structure preserves under external pressure. As is expected, LaTe$_{2}$ experiences a semiconductor–metal–superconductor transition at around 4.5 GPa, and its $T_{\rm c}^{\rm onset}$ monotonously increases to 4.5 K at 20 GPa and remains almost constant. The estimated upper critical field is 7 T, an order of magnitude higher than that of CeTe$_{2}$. Interestingly, the LaTe$_{1.6}$Sb$_{0.4}$ shows a record-high $T_{\rm c}^{\rm onset}$ at 6.5 K in reported RETe$_{2}$ compounds (RE: rare earth elements). Further theoretical analyses reveal that the local vibrational modes of Te(1) and La dominate the spectral weight of electron-phonon coupling. To give a complete picture of the present system, we also calculate its phonon instability and find that the Fermi surface nesting alone is not adequate to explain the observed adjustable CDW ordering. Sample Preparation. Single crystalline LaTe$_{2}$ and LaTe$_{1.6}$Sb$_{0.4}$, with a size of about $2{\,\rm mm} \times 1.5{\,\rm mm} \times 1$ mm, were obtained by chemical vapor transport reactions in quartz tubes using I$_{2}$ as the transporting agent. The starting materials are La ingot (Alfa, 99.99%), Te pieces (Alfa, 99.999%), and Sb grains (Alfa, 99.999%) with stoichiometric ratio of 1 : 2 and 1 : 1.6 : 0.4 for LaTe$_{2}$ and LaTe$_{1.6}$Sb$_{0.4}$, respectively, as the recipe reported earlier.[4] The temperatures of high and low ends are set as 1220 K and 1120 K, respectively. Due to the relative air sensitivity, the obtained crystals were kept in the glove box.
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Fig. 1. (a) Crystal structure of RETe$_{2}$. (b) Optical photograph of LaTe$_{2- x}$Sb$_{x}$ ($x = 0$ and 0.4) single crystals. The length of the grid lines is 2 mm. (c) Atomic force microscope topographic image of LaTe$_{2}$ thin flakes. Step edges with a height of 1 nm (red curve) are shown along the white dashed line. The image scale is 200 nm$^{2}$. (d)–(g) Energy dispersive spectroscopy of LaTe$_{2- x}$Sb$_{x}$ ($x = 0$ and 0.4).
Characterization. Chemical compositions of LaTe$_{2- x}$Sb$_{x}$ single crystals were determined by energy dispersive spectroscopy (EDS) and inductively coupled plasma (ICP) measurements. EDS mapping images were obtained using an ARM-200F (JEOL, Tokyo, Japan). The surface properties of the sample were determined by atomic force microscopy (AFM, Bruker). Selected area electron diffraction was performed by using a high-resolution transmission electron microscopy (HRTEM, Tecnai F20 super-twin). Electrical resistivity ($\rho$) and specific heat ($C_{\rm p}$) were measured through the physical property measurement system (PPMS, Quantum Design). in situ High-Pressure Measurements. In situ high-pressure experiments were carried out in a diamond anvil cell with a culet of 300 µm in diameter. We used the four-wire method to measure the electrical transport properties. Cubic boron nitride (cBN) powders were employed as the pressure medium and insulating material. The single-crystal sample with a size of about 150 µm $\times 150$ µm $\times 10$ µm were put into the boron nitride hole and then attached four gold wires of 20 µm in diameter on top of the sample surface. The pressure was calibrated with the ruby fluorescence method at room temperature each time before and after the measurement. In addition, in situ high-pressure powder x-ray diffraction (PXRD) measurements were performed at beamline BL15U of Shanghai Synchrotron Radiation Facility at wavelength $\lambda = 0.6199$ Å. Theoretical Calculations. Electronic structures of LaTe$_{2}$ were studied with the density functional theory calculations as implemented in the Quantum ESPRESSO (QE) package.[13] Interactions between electrons and nuclei were described by the RRKJ-type ultrasoft pseudopotentials,[14] which were taken from the PSlibrary.[15,16] Generalized gradient approximation of the Perdew–Burke–Ernzerhof formula[17] was adopted for the exchange-correlation functional. A kinetic energy cutoff of the plane-wave basis was set to 80 Ry. A $20 \times 20\times 10$ $\boldsymbol{k}$-point grid was adopted for the Brillouin zone (BZ) sampling. Gaussian smearing method with a width of 0.004 Ry was used for the Fermi surface broadening. In the calculation of the phonon spectrum at ambient pressure, a smearing factor of 0.008 Ry was used. Lattice constants were fixed to the experimentally measured values and only internal atomic positions were optimized with the Broyden–Fletcher–Goldfarb–Shanno (BFGS) quasi-Newton algorithm[18] until the force on all atoms was smaller than 0.0002 Ry/Bohr. Orbital-projected FSs of LaTe$_{2}$ under different pressures were obtained by combining the Vienna ab initio Simulation (VASP) package,[19-21] with the post-processing VASPKIT package[22] and visualized by using the FermiSurfer package.[23] For electronic susceptibility $\chi$, its real and imaginary parts are, respectively, defined as \begin{align*} &\chi'(\boldsymbol{q})=\sum\limits_{nn'\boldsymbol{k}}\frac{f(\varepsilon_{\boldsymbol{k}n})-f(\varepsilon_{\boldsymbol{k} +\boldsymbol{q}n'})}{\varepsilon_{\boldsymbol{k}n}-\varepsilon_{\boldsymbol{k}+\boldsymbol{q}n'}},\notag\\ &\chi''(\boldsymbol{q})=\sum\limits_{nn'\boldsymbol{k}} {\delta (\varepsilon_{\boldsymbol{k}n}-\varepsilon_{\scriptscriptstyle{\rm F}})\delta (\varepsilon_{\boldsymbol{k}+\boldsymbol{q}n'}-\varepsilon_{\scriptscriptstyle{\rm F}})},\notag \end{align*} where $f(\varepsilon_{\boldsymbol{k}n})$ is the Fermi–Dirac distribution function and $\varepsilon_{\boldsymbol{k}n}$ is the energy of band $n$ at vector $\boldsymbol{k}$. Dynamical matrix and electron-phonon coupling (EPC) were calculated within the framework of density functional perturbation theory (DFPT),[24,25] as implemented in QE, and the BZ was sampled with a $4 \times 4\times 2$ $\boldsymbol{q}$-point grid and a $60 \times 60\times 30$ $\boldsymbol{k}$-point mesh, respectively. Results and Discussion. Shiny LaTe$_{2- x}$Sb$_{x}$ ($x = 0$ and 0.4) single crystals were obtained by chemical vapor transport reactions, as shown in Fig. 1(b). Layered LaTe$_{2}$ can be exfoliated into thin flakes, see Fig. 1(c), indicating that the inter-layer interactions of LaTe$_{2}$ are weak. The atomic ratios of La : Te : Sb are determined to be 1 : 1.89 : 0 and 1 : 1.56 : 0.37 by the ICP measurements, which are consistent with the results of EDS analyses in Figs. 1(d)–1(g). The $\rho $–$T$ curves of LaTe$_{2- x}$Sb$_{x}$ ($x = 0$ and 0.4) from 2 to 300 K are plotted in Fig. 2(a). As the temperature decreases, both $\rho (T)$ curves increase rapidly, indicating that the two compounds exhibit semiconducting behaviors. The specific heat $C_{\rm p}$ of LaTe$_{2- x}$Sb$_{x}$ ($x = 0$ and 0.4) below 10 K are measured and we fitted the lowest-temperature data using the equation $C_{p}/T=\gamma +\beta T^{2}$ (5 K$^{2} < T^{2} < 25$ K$^{2}$) in Fig. 2(b), and both values of $\gamma$ are zero ($\pm 1$), in good agreement with semiconducting behaviors. Meanwhile, the Debye temperatures ($\varTheta_{\rm D}$) are calculated by using the relation $\varTheta_{\rm D}$=(12$\pi^{4}n$R/5$\beta)^{1/3}$ with $n = 3$, where $R$ is the molar gas constant. The values of $\varTheta_{\rm D}$ are 165(2) K and 179(2) K, respectively. We further measured the selected area electron-diffraction patterns parallel to the [001] zone axis at room temperature, as shown in Figs. 2(c) and 2(d). As the Sb content increases, the CDW wave vector $\boldsymbol{q}$ changes dramatically from 0.5$a^*$ to 0.73$a^*$, consistent well with the previous report.[9] This result of electron diffraction test reveals that doping by Sb substitution can easily adjust the $\boldsymbol{q}_{\scriptscriptstyle{\rm CDW}}$ in LaTe$_{2- x}$Sb$_{x}$. Here, $a^*$, $b^*$, and $c^*$ represent inverted space wave vectors.
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Fig. 2. (a) Electrical resistivity ($\rho$) as a function of the temperature in LaTe$_{2- x}$Sb$_{x}$ single crystals. Inset: log $\rho$ versus $T^{-1}$ at 10 K-20 K. (b) $T^{2}$ dependent heat capacity $C_{\rm p}/T$ in LaTe$_{2- x}$Sb$_{x}$ single crystals. [(c), (d)] Selected area electron diffraction of LaTe$_{2- x}$Sb$_{x}$ ($x = 0$ and 0.4) taken at room temperature along the [001] axis. Strong ($hk$0) reflections with $h+k$ even correspond to the sublattice. The CDW wave vectors are $q = 0.50a^*$ and 0.73$a^*$, respectively. The weak unlabeled peaks are due to the presence of a CDW.
Figure 3 shows the in situ PXRD patterns of LaTe$_{2}$ collected at room temperature. At $P = 0.2$ GPa, all of the diffraction peaks can be well indexed as the space group of $P4/nmm$. The refinements based on the reported structure[4] converge to $R_{\rm p} = 1.10{\%}$ and $R_{\rm wp} = 1.58{\%}$, see Fig. 3(a). In Fig. 3(b), a monotonic shift of all diffraction peaks is observed with increasing external pressure, indicating the shrinkage of the unit cell. No additional peaks and phase transitions emerge below 34.8 GPa. The PXRD pattern of the sample after releasing pressure is almost the same as the pristine one. It indicates that LaTe$_{2}$ does not decompose at high pressure. The results of Rietveld refinements for all patterns are summarized in Table S1. Therefore, no structure transition can be observed in pressurized LaTe$_{2}$ up to 34.8 GPa. Next, we extracted the lattice constants of $a$, $c$, and volume of the unit cell ($V$), and summarized them in Figs. 3(c)–3(e). The $a$, $c$, and $V$ decrease smoothly and continuously with the increase of pressure. The whole contraction ratio up to $P = 35$ GPa of $(a_{0}-a)\times 100{\%}/a_{0}=6.85{\%}$, $(c_{0}-c)\times 100{\%}/c_{0}=7.75{\%}$, $(V_{0}-V)\times 100{\%}/V_{0}=19.96{\%}$, where $a_{0}$, $c_{0}$, and $V_{0}$ are the values under ambient pressure. Meanwhile, we fitted the pressure-dependent $V$ using the third-order Birch–Murnaghan equation[26] \begin{align*} P=\,&\frac{3}{2}B[(V_{0}/V)^{7/3}-(V_{0}/V)^{5/3}]\notag\\ &\cdot\Big\{1+\frac{3}{4}(B'-4)[(V_{0}/V)^{2/3}-1]\Big\}, \end{align*} where $B$ is the isothermal bulk modulus; $B'$ the first pressure derivative of $B$; $V$ and $V_{0}$ are the high-pressure volume and zero-pressure volume, respectively. The fitting curve is plotted as red solid line in Fig. 3(e). The obtained $B$ of LaTe$_{2}$ is 95(4) GPa, which is higher than 2$H_{\rm c}$-MoS$_{2}$ (50 GPa) and lower than 2$H_{\rm a}$-MoS$_{2}$ (110 GPa) and IrTe$_{2}$ (126 GPa).[27,28]
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Fig. 3. (a) Rietveld refinements of in situ PXRD pattern of LaTe$_{2}$ collected at 0.2 GPa. (b) PXRD patterns of LaTe$_{2}$ measured under different pressures. [(c), (d), (e)] Pressure dependence of the lattice parameters $a$, $c$, and $V$ of LaTe$_{2}$. The red line in (e) is a fitted curve of the Birch–Murnaghan equation.
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Fig. 4. [(a), (b)] $R$–$T$ curves of 2–300 K and 2–7 K for LaTe$_{2}$ under different pressures. (c) Temperature dependence of the upper critical fields $\mu_{0}H_{\rm c 2}$ at different pressures of LaTe$_{2}$. Solid lines are the Ginzburg–Landau fitting curves, $H=H_{0}[1-(T/T_{\rm c})^{2}]$. The inset is $R/R_{\rm 5K}$ of 14.3 GPa around $T_{\rm c}$ under an external magnetic field along the $c$ direction. (d) Hall coefficients of LaTe$_{2}$. Inset shows the carrier density (related to Hall coefficient) versus pressure. [(e), (f)] $R$–$T$ curves of 2–300 K and 2–7 K for LaTe$_{1.6}$Sb$_{0.4}$ under different pressures.
Figure 4(a) shows the temperature dependence of resistance ($R$–$T$ curve) of LaTe$_{2}$ at various pressures. At low pressures ($ < 4.6$ GPa), the $R$ increases rapidly upon cooling. It shows typical semiconducting behavior, which is quickly suppressed by increasing pressure. Above 4.6 GPa, the LaTe$_{2}$ becomes metal, and the values of $R$ at 300 K decrease by three orders of magnitude. It demonstrates that LaTe$_{2}$ undergoes a semiconductor-metal transition. Meanwhile, a sudden drop of $R$ at 2.8 K is observed at $P = 4.6$ GPa, indicating the onset of a superconducting transition. By increasing the pressure to 20.2 GPa, the $T_{\rm c}^{\rm onset}$ reaches a maximum of 4.6 K. Upon further increasing the pressure, the $T_{\rm c}^{\rm onset}$ slightly decreases, as shown in Fig. 4(b). The resistivity versus temperature at different pressures of LaTe$_{2}$ are shown in the insets of Fig. 4(c) and Fig. S1 in the Supporting Information. All the superconducting transitions are gradually suppressed to lower temperature with increasing $H$. The superconductivity completely disappears above 2 K under 6 T. The pressure dependence of $\mu_{0}H_{\rm c 2}$(0) is plotted in Fig. 4(c). Interestingly, the zero-temperature upper critical fields $\mu_{0}H_{\rm c 2}$(0) of 9.1 GPa and 14.3 GPa are estimated to be 7.2 T and 7.6 T, respectively, which are slightly larger than their Pauli-limited critical fields 6.4 T and 7.1 T, respectively.[29] We measured the pressure-dependent $R_{\scriptscriptstyle{\rm H}}$ and carrier density $n$ of LaTe$_{2}$ at 100 K, shown in Fig. 4(d), with the raw data presented in Fig. S2. The values of $R_{\scriptscriptstyle{\rm H}}$ are positive below 1.3 GPa and then turn to negative above 4.1 GPa, revealing the dominant carrier changes from hole to electron. This result roughly consist with the semiconductor–metal–superconductivity transition. Furthermore, the estimated $n$ gradually increases from $10^{20}$ cm$^{-3}$ at 0.8 GPa to $10^{23}$ cm$^{-3}$ at 34.4 GPa, which supports the transition of semiconductor to metal.[30-32] Figures 4(e)–4(f) show the $R$–$T$ curve of LaTe$_{1.6}$Sb$_{0.4}$ at different pressures. The transitions of semiconductor-to-metal and superconducting of LaTe$_{1.6}$Sb$_{0.4}$ are also observed. Superconductivity emerges at $P = 2.5$ GPa, and the $T_{\rm c}^{\rm onset}$ increases to the maximal value of 6.5 K. The $R_{\scriptscriptstyle{\rm H}}$ of LaTe$_{1.6}$Sb$_{0.4}$ is similar to LaTe$_{2}$, as shown in Figs. S3 and S4 in the Supporting Information. Accordingly, we summarize all transport results and show a pressure dependence superconducting phase diagram of LaTe$_{2- x}$Sb$_{x}$ ($x = 0$ and 0.4) in Fig. 5. With increasing pressure, both LaTe$_{2- x}$Sb$_{x}$ ($x = 0$ and 0.4) undergo semiconductor–metal–superconductivity transition. The highest $T_{\rm c}^{\rm onset}$ increases from 4.6 to 6.5 K with the doping of Sb. The effect of Sb substitution will be discussed later.
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Fig. 5. Pressure-dependent superconducting phase diagram of LaTe$_{2- x}$Sb$_{x}$ ($x = 0$ and 0.4). The $T_{\rm c}$ at each pressure is defined as the temperature at which the resistivity drops by an order of magnitude with respect to the resistivity just above the superconducting transition.
The CDW transition in the compound usually occurs when a soft phonon mode at a specific wave vector $\boldsymbol{q}$ emerges. The calculated phonon spectrum of LaTe$_{2}$ at the ambient pressure is shown in Fig. 6(a). There is an imaginary phonon mode near the $X$ point (${\boldsymbol q} \sim 0.5a^*$), in good agreement with the experimentally observed CDW wave vector ($\boldsymbol{q} _{\scriptscriptstyle{\rm CDW}} \sim 0.5a^*$). The corresponding Fermi surface cut in the $c^*=0$ plane is displayed in Fig. 6(b). There are several quasi-2D Fermi surface sheets across the BZ interior and a small Fermi pocket around the $\varGamma$ point. Based on the information of the Fermi surface, we calculated the electronic susceptibility $\chi (\boldsymbol{q)}$. The maximum peak in the real part $\chi'(\boldsymbol{q})$ appears around the $X$ point ($\boldsymbol{q} \sim 0.5a^*$), see Fig. 6(c), suggesting the instability of the electronic system.[33] On the other hand, the imaginary part $\chi''(\boldsymbol{q})$ reflects the information of FS nesting.[34] From Fig. 6(d), it can be seen that the strongest peak in $\chi''(\boldsymbol{q})$ locates at the middle point of the $\varGamma$–$M$ path, indicating that the nesting vector $\boldsymbol{q}_{N}$ is around (0.25$a^*$, 0.25$b^*$). In addition, weaker peaks of $\chi''(\boldsymbol{q})$ show up at the CDW wave vector $\boldsymbol{q}_{\mathrm{CDW}} = 0.5a^*$. There is also a hot spot at the $\varGamma$ point, which comes from the intraband contribution of a weakly dispersing band, irrelevant to the FS nesting.[35] These facts suggest that the FS nesting alone cannot induce CDW instability, and the formation of CDW must be assisted by the EPC interaction.
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Fig. 6. (a) Phonon spectrum of LaTe$_{2}$ at ambient pressure. The size of the red dot represents the phonon linewidth $\gamma_{q\nu}$ of the three acoustic branches. (b) Calculated Fermi surface cut in the $k_{x}$–$k_{y}$ plane. Color scales on the Fermi surface represent the Fermi velocities, where the red and blue correspond to the highest and zero values, respectively. The real part (c) and imaginary part (d) of the electron susceptibility $\chi$ as a function of ($q_{x}$, $q_{y}$) with $q_{z} = 0$ for LaTe$_{2}$, where the red and blue colors correspond to the highest and lowest values, respectively.
To explore the contribution of EPC to the CDW formation, we further calculated the phonon linewidth $\gamma$, which is defined as \begin{align*} \gamma_{q\nu}=\,&2\pi \omega_{q\nu}\sum\limits_{ij} \int \frac{d^{3}k}{\varOmega _{\scriptscriptstyle{\rm BZ}}}|g_{q\nu}(k,i,j)|^{2}\notag\\ &\times \delta (\varepsilon_{q,i}-\varepsilon_{\scriptscriptstyle{\rm F}})\delta (\varepsilon_{k+q,j}-\varepsilon_{\scriptscriptstyle{\rm F}}),\notag \end{align*} where $g_{q\nu}(k,i,j)$ is the EPC matrix element. The phonon linewidth $\gamma$ takes into account both the effects of the momentum-dependent EPC and the FS nesting. The information of $\gamma$ is overlaid on the phonon spectrum in the form of red dots in Fig. 6(a), whose sizes are proportional to their values. It can be seen that there is indeed considerable EPC strength at $\boldsymbol{q}_{\mathrm{CDW}}$. Therefore, we conclude that the effect of FS, the strong EPC, and the imaginary phonon mode associated with the structural distortions play indispensable roles in inducing CDW instability.
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Fig. 7. (a) Phonon spectrum of LaTe$_{2}$ under 4.5 GPa. The size of the red dot represents the EPC strength $\lambda_{q\nu}$. (b) Total and projected phonon density of states (PHDOS). (c) Orbital-resolved electronic band structure of LaTe$_{2}$ under 4.5 GPa. The weights of the main orbitals are displayed. The right panel shows the total and partial density of states. (d) Eliashberg spectral function $\alpha^{2}F(\omega)$ (blue line) and integrated EPC constant $\lambda (\omega)$ (red dashed line).
As we know, the CDW state competes or coexists with other electronically ordered states like superconductivity.[36-38] In the CDW-bearing LaTe$_{2}$, we find experimentally that with the increase of pressure, the superconductivity emerges at $P = 4.5$ GPa. To explore the superconducting mechanism, we carried out the first-principles calculations on the lattice dynamics, electronic structure, and EPC strength of LaTe$_{2}$ at 4.5 GPa. The phonon spectrum of LaTe$_{2}$ at 4.5 GPa is plotted in Fig. 7(a). We find that there is no imaginary phonon mode in the whole BZ, indicating that the CDW has been completely suppressed at 4.5 GPa. This result agrees with the sign change of $R_{\scriptscriptstyle{\rm H}}$ and $n$. The corresponding projected phonon DOS (PHDOS) is shown in Fig. 7(b). It can be seen that the low-frequency bands are dominated by the vibrations of Te(1) and La atoms, while the high-frequency modes mainly come from Te(2) and La vibrations. Figure 7(c) shows the orbital-resolved electronic band structure of LaTe$_{2}$ under 4.5 GPa. One can see that two bands cross the Fermi level $E_{\rm F}$ from $-1$ eV to 1 eV, indicative of metallic behavior. Based on the orbital analysis, these bands are mainly contributed by the 5$p$ orbitals of Te(1) and 5$d$ orbitals of La. The contribution of Te(1) 5$p$ is larger than that of La 5$d$. From the calculated Fermi surface of LaTe$_{2}$ under different pressures, as shown in Fig. 8, we find that there is a charge transfer from the Te(1) 5$p$ to the La 5$d$ state and the spherical Fermi pocket dominated by La 5$d$ states at the BZ center enlarges with the applied pressure. From the momentum- and mode-resolved EPC parameter $\lambda_{q\nu}$ [as indicated by the size of red dots in Fig. 7(a)], we find that the largest contribution to the EPC comes from the acoustic phonon branch, which corresponds to the sharp peak around 50 cm$^{-1}$ in $\alpha^{2}F(\omega)$ [Fig. 7(d)]. The calculated total EPC constant $\lambda$ of LaTe$_{2}$ at 4.5 GPa is 0.717 and the calculated $T_{\rm c}$ is about 2.97 K, which is in good accordance with the measured value. This means that the superconductivity in LaTe$_{2}$ under pressure can be explained in the framework of the Bardeen–Cooper–Schrieffer theory. As to superconductivity in LaTe$_{1.6}$Sb$_{0.4}$, it is known that the Fermi-surface-driven CDW is stable over a large range of hole doping in the LaTe$_{2-x}$Sb$_{x}$ system.[9] Sb substitution removes electrons from the Te square-sheet bands and changes the CDW nesting vector $\boldsymbol{q}$ changing from 0.50$a^*$ to 0.73$a^*$. Here, the superconductivity still emerges in the proximity of CDW with a higher $T_{\rm c}$, considering the decreased Debye temperature, we speculate that the introduced Sb greatly improves the electron-phonon coupling strength of the system. We find no evidence for disorder in the LaTe$_{1.6}$Sb$_{0.4}$ crystal.
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Fig. 8. (a) The Brillouin zone of LaTe$_{2}$. The high-symmetry paths in the BZ are indicated by the red lines. The Fermi surfaces of LaTe$_{2}$ at (b) 0 GPa, (c) 4.5 GPa and (d) 22.5 GPa, respectively. The yellow and blue colors reflect to the weights of Te 5$p$ and La 5$p$ orbitals, respectively.
The discovery of pressure-induced superconductivity in LaTe$_{2}$ has led to a deeper understanding of the origin of superconductivity in the RETe$_{2}$ system. Firstly, both LaTe$_{2}$ and CeTe$_{2}$ undergo semiconductor–metal–superconductivity transitions under pressure. The superconductivity of the former appears at 4.6 GPa with a maximum $T_{\rm c}^{\rm onset}$ of 4.5 K. The latter superconductivity occurs at 0.1 GPa and is more sensitive to pressure. It is noteworthy that, with the increasing pressure, the $T_{\rm c}$ of CeTe$_{2}$ increases with the increase of the magnetic phase transition temperature.[6] Secondly, according to the previous calculation of electronic structure, it is suggested that increased self-doped Te(1) 5$p$ hole carriers are responsible for the pressure-induced superconductivity in CeTe$_{2}$, but the complex magnetic orders make the superconductivity mechanism still difficult to understand.[8] The calculation $T_{\rm c}$ of LaTe$_{2}$ under pressure in this work is in good accordance with the measured value, which can suggest that the electron-phonon coupling is responsible for pressure-induced SC in RETe$_{2}$ systems, rather than magnetism. Thirdly, by doping Sb to adjust $\boldsymbol{q}_{\scriptscriptstyle{\rm CDW}}$, the $T_{\rm c}^{\rm onset}$ of LaTe$_{2}$ can be enhanced to 6.5 K, which is the maximum superconducting transition temperature in the RETe$_{2}$ system. In summary, we have studied the in situ high-pressure electrical resistivity measurements up to 40 GPa in LaTe$_{2- x}$Sb$_{x}$ ($x = 0$ and 0.4), together with the high-pressure x-ray diffraction and first-principles calculations. Without any involvement of 4$f$ electrons, superconductivity can be feasibly realized in LaTe$_{2}$ with a doubled $T_{\rm c}$ than that of CeTe$_{2}$. An interesting discovery is that the $T_{\rm c}^{\rm onset}$ can be further raised to 6.5 K by introducing surplus electrons to the system by Sb substitution. Combined with theoretical calculations, our experimental observations can be well explained by the scenario of charge transfer from Te(1) 4$p$ orbitals to empty La 5$d$ orbitals. The same argument should also be applicable to the reported semiconductor–metal–superconductor transition in CeTe$_{2}$. Further calculations of charge susceptibility and phonon dispersion demonstrate that the origin of CDW is not only determined by Fermi surface nesting but also by appreciable EPC contributions. Meanwhile, EPC is dominated by the low-frequency phonon modes of Te(1) and La atoms. This work highlights the importance of non-empty 5$d$ orbitals to the emergence of superconductivity in the RETe$_{2}$ system. Acknowledgment. This work was financially supported by the National Key Research and Development Program of China (Grant Nos. 2018YFE0202600, 2021YFA1401800, and 2017YFA0304700), the National Natural Science Foundation of China (Grant Nos. 51922105, 11804184, 11974208, 11774424, 12174443, U1932217, and 11974246), and Beijing Natural Science Foundation (Grant No. Z200005). The HP-XRD was performed at the beamline BL15U1, Shanghai Radiation Facility (SSRF). Computational resources were provided by the Physical Laboratory of High Performance Computing at Renmin University of China. This work was also supported by the Synergetic Extreme Condition User Facility (SECUF). 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